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Divisibility Tests - Lesson Year 7 Maths

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Dr Frost Learning is a registered charity in England and Wales (no 1194954) www.drfrost.org @DrFrostMaths Last modified: Contact the resource team: resources@drfrost.org @DrFrostResource Jasmine Vincent Divisibility Tests 7th January 2025
How to use these slides Dr Frost Learning is a registered charity in England and Wales (no 1194954) Slide Title Explanation Default Animations* Recap To be used as a prior knowledge check or to review prerequisite knowledge. Can be used as a starter or as part of the main lesson. Green click-to-reveal boxes. The Big Idea To be used to highlight key concepts or theorems. This could include the ‘why’ of the topic - including “real-life” contextual scenarios, or putting into context of other mathematical concepts (past and future). Usually in sequence with some green click-to-reveal boxes. Example To be modelled by the teacher. Solution animates in sequence. Test Your Understanding To be completed by students and used for Assessment for Learning, primarily using mini-whiteboards. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final answer to reveal full solution. Example Problem Pair To be used as ‘Example’ &‘Test Your Understanding’ above, within the same slide to provide scaffold via visible modelled solution. TYU column is blank initially, to focus attention on example. Reveal question by clicking ‘Test Your Understanding’ banner. Example animates in sequence. Click the header to reveal TYU question, then green click-to- reveal boxes. Quickfire Questions To be used as fluency practice. Multiple questions in rapid succession, for calculations that can be completed mentally. Often used for shorter questions/ formulae or to isolate a small part of the method. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final line to reveal full solution. Multi-choice Question To be used as a diagnostic question. Multiple choice questions, with plausible distractors, to allow teachers to diagnose misconceptions and errors in student thinking, then adapt their lesson accordingly. Arrow points to answer, on click. Exam Question To be completed by teacher or student. Green click-to-reveal boxes. Though many slides in this resource will have titles specific to the topic, the slide titles in the table below are used consistently within DFL resources for specific pedagogical purposes. Any atypical use of a slide type, including any change of animation* or intended use, will be outlined in the Teacher Notes for the slide. How to use these slides Dr Frost Learning is a registered charity in England and Wales (no 1194954) Slide Title Explanation Default Animations* Recap To be used as a prior knowledge check or to review prerequisite knowledge. Can be used as a starter or as part of the main lesson. Green click-to-reveal boxes. The Big Idea To be used to highlight key concepts or theorems. This could include the ‘why’ of the topic - including “real-life” contextual scenarios, or putting into context of other mathematical concepts (past and future). Usually in sequence with some green click-to-reveal boxes. Example To be modelled by the teacher. Solution animates in sequence. Test Your Understanding To be completed by students and used for Assessment for Learning, primarily using mini-whiteboards. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final answer to reveal full solution. Example Problem Pair To be used as ‘Example’ &‘Test Your Understanding’ above, within the same slide to provide scaffold via visible modelled solution. TYU column is blank initially, to focus attention on example. Example animates in sequence, followed by TYU question with green click-to-reveal boxes for solution steps. Quickfire Questions To be used as fluency practice. Multiple questions in rapid succession, for calculations that can be completed mentally. Often used for shorter questions/ formulae or to isolate a small part of the method. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final line to reveal full solution. Multi-choice Question To be used as a diagnostic question. Multiple choice questions, with plausible distractors, to allow teachers to diagnose misconceptions and errors in student thinking, then adapt their lesson accordingly. Arrow points to answer, on click. Exam Question To be completed by teacher or student. Green click-to-reveal boxes. Though many slides in this resource will have titles specific to the topic, the slide titles in the table below are used consistently within DFL resources for specific pedagogical purposes. Any atypical use of a slide type, including any change of animation* or intended use, will be outlined in the Teacher Notes for the slide.
Teacher Notes Dr Frost Learning is a registered charity in England and Wales (no 1194954) Key Points Solution step – click to reveal Question/Discussion Prompt Key: ! To be written in books All slides include pedagogical detail in the ‘Notes’ section for each slide. Throughout the slides, this symbol refers to a web link. Unless otherwise specified, this will be to some functionality within DF. Future Links • Simplifying fractions • Identifying properties of a number based on it’s prime factorisation Prerequisite Knowledge • Identify multiples and factors • Find all factor pairs of a number • Identify common factors • Identify common multiples • Identify prime numbers • Prime factorisation
Dr Frost Learning is a registered charity in England and Wales (no 1194954) Dr Frost Learning is a registered charity in England and Wales (no 1194954) Using the Dr Frost online platform Skills in this Lesson 163 Divisibility laws from 3 to 11 and dealing with larger divisors 163a Know if a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10. 163b Know if a number is divisible by 11. 163c Determine an unknown digit to make a number divisible by 3, 4, 6, 8 or 9. 163d Know if a number is divisible by a large number by combining divisibility rules. TEACHERS Generate a random worksheet involving skills in this PowerPoint (for printing or online task setting). STUDENTS Start an independent practice involving skills in this PowerPoint. drfrost.org/w/866 Clicking this box takes you to a single question practice for a subskill to allow you further Test Your Understanding opportunities. (e.g. drfrost.org/s/123a) drfrost.org/p/866 drfrost.org/s/123a
Dr Frost Learning is a registered charity in England and Wales (no 1194954) Dr Frost Learning is a registered charity in England and Wales (no 1194954) Contents Prerequisite Check: Factors, Multiples and Primes Divisibility laws: and Divisibility laws: and Divisibility laws: and Divisibility laws: and Prime Numbers Missing digits Combining divisibility laws Exercise For lessons covering many concepts, please click the below to navigate quickly to the relevant part of the lesson.
Prerequisite Check: Factors, Multiples and Primes 1 Write as a product of prime factors. 𝟏𝟖𝟎=𝟐𝟐 ×𝟑𝟐 ×𝟓 2 List the first multiples of 𝟖,𝟏𝟔,𝟐𝟒,𝟑𝟐,𝟒𝟎 3 List all the factors of 𝟏,𝟐,𝟑,𝟒,𝟔,𝟗,𝟏𝟐,𝟏𝟖,𝟑𝟔 4 5 Which of the numbers below are even? and 6 What is the smallest prime number? 𝟐 7 Is a prime or composite number? so is divisible by and making it a composite number 8 Identify the HCF of and . 𝟏𝟐 9 Identify the LCM of and . ? ? ? ? ? ? ? ? 10 Identify the HCF and LCM of and . HCF: LCM: ? Is a prime number? Explain why or why not. Yes, is a prime number because it has exactly factors; and itself. ? Show all solutions
The Big Idea: Divisibility Rules Mrs Clark How do you know if a number is divisible by ? It will end in a 0, 2, 4, 6 or 8 It will be an even number Both correct! How do you know if a number is divisible by ? It will end in a … … or a ! Exactly! 163a drfrost.org/s/
The Big Idea: Divisibility Rules Mrs Clark Do you know any other divisibility rules already? Factor Rule Last digit is even or Last digit is or Logan Any number that ends in a is divisible by Great! Let’s see if we can fill in the rest of the table Last digit is 163a drfrost.org/s/
is divisible by and ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by and/or ? Yes, because it ends in an even digit. No, because it doesn’t end in a or a . Yes, because it ends in a . No, because it doesn’t end in a . Yes, because it ends in an even digit. ? ? ? ? ?
The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark We can use divisibility laws for other numbers too! Some are more complicated than others… Nathan How am I meant to know if is divisible by something other than , or ? Factor Rule Last digit is even or Last digit is or Last digit is
Do you notice anything special about the highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Number Calculation Digit Sum Nathan They are all numbers in the times table; and their digit sum is also in the times table A number is divisible by if the sum of its digits if divisible by Is divisible by ? The digit sum of is: is divisible by therefore is divisible by
Number Calculation Digit Sum Do you notice anything special about these highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Logan I’ve noticed that the numbers in the times table have a digit sum of Will this always be true?
The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the times table in more detail Number Calculation Digit Sum Logan doesn’t fit the pattern; maybe the rule is the digits sum to either or ? Nathan You should test it out on a bigger number you know is divisible by – like Number Calculation Digit Sum is also in the times table; so the digit sum just needs to be divisible by as well! A number is divisible by if the sum of its digits if divisible by
is not divisible by or therefore is not divisible by or ? is divisible by therefore is divisible by . is divisible by therefore is divisible by is not divisible by therefore is not divisible by . is divisible by and therefore is divisible by both and . is divisible by but not by , therefore is divisible by but not by ? ? ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by , or both? Is divisible by , or both? Is divisible by or both?
The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is 163a drfrost.org/s/
Example: Checking divisibility by or Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Is the number divisible by or ? The divisibly rules we’ve seen so far either use the last digit or the digit sum. The last digit of is . This means that is divisible by , but not or . The digit sum is: is divisible by and therefore is divisible by and . is divisible by and .
Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Divisible by                  Tick the factor(s) that each number is divisible by ? ? ? ? ? ? ? Make a -digit number that matches each statement A multiple of Largest possible multiple of Multiple of and ? ? ? ? ? 163a drfrost.org/s/ 1 2 a b c Show all solutions
The Big Idea: Divisibility Rules Mrs Clark All these numbers are divisible by ; do you notice anything special about them? 163a drfrost.org/s/ 216 10524 27340 99048 459136 It might help to look specifically at the last digits of each number Hannah If we treat the last digits as a -digit number; they are all multiples of A number is divisible by if the number formed by the last digits is divisible by Can you explain why this works?
The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the number 27340 27300 40 can be split into and is divisible by – we can tell this because it ends in two zeros Any number that ends in two zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by is divisible by therefore we know that is divisible by
Example: Divisibility by 163a drfrost.org/s/ 12345 12300 45 First; we split the into a multiple of and the rest of the number We know is divisible by because it is a multiple of is not divisible by therefore we know that is not divisible by Is the number divisible by ?
is divisible by is also divisible by Therefore is divisible by . is divisible by is not divisible by Therefore is not divisible by . is divisible by is divisible by Therefore is divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The digit sum is a multiple of Last digit is 163a drfrost.org/s/
The Big Idea: Divisibility Rules 163a drfrost.org/s/ True or False? All multiples of are even… True or False? All multiples of are even… True or False? All multiples of are even… False True True False False True Only some multiples of are even 3,6,9,12,15,18,21,24 ,27 ,30… Viktor The multiples of that are even are also multiples of ; and all the even numbers are multiples of A number is divisible by if it divisible by both and
Example: Divisibility by Is the number divisible by ? ends in an even number therefore it is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by both and therefore is divisible by A number is divisible by if it divisible by both and Is the number divisible by ? ends in an even number therefore it is divisible by The digit sum is: is not divisible by therefore is not divisible by is divisible by but not by therefore is not divisible by
ends in an even digit therefore is divisible by The digit sum is: is not divisible by therefore is not divisible by is not divisible by ends in an even digit therefore is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by ends in an odd digit therefore is not divisible by We don’t need to check if the number is divisible by as well is not divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The digit sum is a multiple of Last digit is 163a drfrost.org/s/ Logan which is why this works… maybe there are other divisibility rules we can find that work in this way?
The Big Idea: Divisibility Rules Mrs Clark Can you remember the divisibility rule for ? 163a drfrost.org/s/ A number is divisible by if the number formed by the last digits of the number is divisible by You can test if a number is divisible by in a similar way; this time we must look at the last digits to see if it is divisible by 23,416 Is divisible by ? If is divisible by then we will know the whole number is divisible by Viktor But how do we know if is divisible by ? To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer is divisible by , therefore is divisible by
FYI: For Your Interest To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer For larger numbers we can use a similar divisibility rule for as we used for . 23 416 23000 416 can be split into and is divisible by – we can tell this because it ends in three zeros Any number that ends in three zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by is divisible by , we can tell by halving the number times (this works because Therefore is divisible by
is divisible by is not an integer therefore is not divisible by Therefore is not divisible by . is divisible by is an integer therefore is divisible by Therefore is divisible by . is divisible by is an integer therefore is divisible by Therefore is divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The last digits form a number divisible by The digit sum is a multiple of Last digit is 163a drfrost.org/s/ Divisible by                      Tick the factor(s) that each number is divisible by ? ? ? ? ? ? Are the statements below always, sometimes or never true? If a number is a multiple of it is also a multiple of If a number is a multiple of it is also a multiple of Always true Sometimes true ? ? 1 2 a b Show all solutions
Example Test Your Understanding Mrs Clark The divisibility rules for and are slightly less intuitive A number is divisible by if when adding and subtracting its digits in an alternative pattern it makes a number divisible by . Is divisible by ? is divisible by therefore is divisible by Is divisible by ? drfrost.org/s/ 163b Is divisible by ? is divisible by (and any number except ) therefore is divisible by is not divisible by therefore is not divisible by ?
Example Test Your Understanding Mrs Clark The divisibility rules for and are slightly less intuitive A number is divisible by if when you double the last digit and subtract it from the rest of the number, the result is divisible by . Is divisible by ? Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by Is divisible by ? Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by drfrost.org/s/ 163b ?
Example Test Your Understanding Double the last digit: Number made by other digits: Subtract: Now we need to check Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by ? A number is divisible by if when you double the last digit and subtract it from the rest of the number, the result is divisible by . Is divisible by ? Double the last digit: Number made by other digits: Subtract: Now we need to check using the same process Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by Is divisible by ? drfrost.org/s/ 163b
The Big Idea: Divisibility Rules Factor Rule Last digit is even or The digit sum is a multiple of The last 2 digits form a number divisible by 4 Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Here’s a summary of all the divisibility rules you have learned:
Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by 7 The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Divisible by                    Tick the factor(s) that each number is divisible by ? ? ? ? ? ? 163a drfrost.org/s/ Show all solutions 163b
Apart from the obvious instant checks (divisibility by , ), we usually only need to mentally check and to have a good ‘guess’ that a number is prime. Is it prime?  No Yes  No Yes  No Yes No! () N What is the largest prime factor we need to test before being certain a number is prime? We can use the square root of the number we are testing; for example so we would need to check up to 14. All composite numbers have a factor (other than 1) up to the square root. ? ? The Big Idea: Prime Numbers ? ? ? ? ? ? ? ? ? ? ? ? ? 163b drfrost.org/s/
Example Test Your Understanding Find which possible digit(s) could go in the box to make  divisible by . Recall the divisibility law for A number is divisible by if it divisible by both and For the number to be divisible by it would need to end in a or For the number to divisible by the digits needs to add up to a multiple of The digit sum so far is… This is already a multiple of therefore the digit in the box also needs to be a multiple of The missing digit could be or Find which possible digit(s) could go in the box to make  divisible by . Recall the divisibility law for The two digit number  needs to be divisible by Consider the numbers in the times table around the number … … The missing digit could be or A number is divisible by if the number formed by the last digits of the number is divisible by drfrost.org/s/ 163c ?
For the number to be divisible by it needs to be divisible by and The number is already divisible by as it ends in a Digit sum: The digit sum is already a multiple of therefore the missing digit must also be divisible by : it could be or For the number to be divisible by the last digits need to make a multiple of The last digits could be: or Out of this list only some are multiples of : the missing digit could be or For the number to be divisible by the last digits needs to make a multiple of Thinking of the times table… …… The missing digit must be Test Your Understanding Find which possible digit(s) could go in the box to make divisible by . Find which possible digit(s) could go in the box to make divisible by . ? Find which possible digit(s) could go in the box to make divisible by . For the number to be divisible by the digit sum needs to be a multiple of Digit sum: To make the digit sum a multiple of it would need to sum to : the missing digit must be Find which possible digit(s) could go in the box to make divisible by . ? ? ? 163c drfrost.org/s/ 1 2 3 4
If we want to check if a number is divisible by , we can show it is divisible by and , they are coprime and have a product of Are these statements true or false? If we want to show that a number is divisible by , we can show it is divisible by and . If we want to show that a number is divisible by , we can show it is divisible by and . The Big Idea: Combining Divisibility Rules False True True False 24 48 7 2 9 6 120 This is sometimes true; all of the multiples of are divisible by and . However, there are common multiples of and that are not divisible by . E.g. is divisible by and but not by . 36 We need to pick two numbers which are coprime, i.e. do not share any factors. Therefore, how should we test if a number is divisible by ? ?
Quickfire Questions What divisibility rules would we use to test divisibility by: and and and and and and and and 163d drfrost.org/s/ and and
Example Test Your Understanding Find which possible digit(s) could go in the box to make divisible by . For the number to be divisible by it must be divisible by and Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or Divisible by Last digits need to be a multiple of : multiples of : , , , , Digits that fit both rules and can therefore be the missing digit: or Find which possible digit(s) could go in the box to make divisible by For the number to be divisible by it must be divisible by and Divisible by Must end in a or Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or The only digit that fits both rules and is therefore the missing digit: drfrost.org/s/ 163d ? ? ? ?
Exercise (Available as a separate worksheet) Show all solutions 1 Tick the factor(s) that each number is divisible by Divisible by Prime                                     ? ? ? ? ? ? [JMC 2011 Q2] How many of the integers are multiples of ? [ ] [ ] [ ] [ ] [ ] Answer: (all numbers have a digit sum that is a multiple of ) 2 [JMC 2004 Q2] Which of the following numbers is exactly divisible by ? 3 Answer: ? ? ?
The number needs to be divisible by and ; the digit sum is and needs to be a multiple of Answer: [JMC 2016 Q11] Which of the following statements is false? [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [JMC 2003 Q13] was a prime year, since is a prime number. In the following ten years there was just one prime year. Which was it? Hint: Use your divisibility rules! [ ] [ ] [ ] [ ] [ ] Answer: is divisible by is divisible by and is divisible by is divisible by [JMC 1999 Q17] The -digit number is a multiple of . Which digit is represented by ? Answer: Therefore which creates a final sum of Exercise (Available as a separate worksheet) Show all solutions 4 5 6 Answer: ( is not a multiple of ) 7 8 ? ? Find which possible digit(s) could go in the box to make divisible by . Current digit sum is ; this is already a multiple of Answer: or ? Find which possible digit(s) could go in the box to make divisible by ? ?
[JMC 2000 Q17] The first and third digits of the five- digit number are the same. If the number is exactly divisible by , what is the sum of its five digits? Answer: Digit sum: must be an odd number between and , and the only odd multiple of in this interval is . [JMC 1997 Q20] A four-digit number was written on a piece of paper. The last two digits were then blotted out (as shown). If the complete number is exactly divisible by three, by four, and by five, what is the sum of the two missing digits? Answer: The missing digits are and [JMC 2012 Q23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number . How many of the numbers on Peter’s list are prime? [ ] [ ] [ ] [ ] [ ] Answer: [Pink Kangaroo 2020 Q16] The digits from to are randomly arranged to make a -digit number. What is the probability that the resulting number is divisible by ? Answer: Exercise (Available as a separate worksheet) Show all solutions 9 ? ? ? ? 10 11 12 8 6
[SMC 2012 Q6] What is the sum of the digits of the largest -digit palindromic number which is divisible by ? Palindromic numbers read the same backwards and forwards, e.g. . Answer: For the number to be divisible by it eithers ends in a or a . It cannot end in a as the first digit cannot be . Therefore, the number is 5**5 The digit sum needs to be a multiple of , this is true if * or therefore the largest is when * which makes the digit sum Exercise (Available as a separate worksheet) Show all solutions 13 The letters , and stand for non-zero digits. The integer ‘’ is a multiple of ; the integer ‘’ is a multiple of ; and the integer ‘’ is a multiple of . What is the integer ‘’? Answer: • is a multiple of , therefore is a multiple of • is a multiple of , the digit sum for would be the same therefore is a multiple of and • Hence is a multiple of • is a multiple of ; it must be a multiple of and as the digits are non-zero and is a multiple of • The digit multiples of that start with a () are: and • Only has a as a multiple of • Therefore and N ? ?

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Divisibility Tests - Lesson Year 7 Maths

  • 1.
    Dr Frost Learningis a registered charity in England and Wales (no 1194954) www.drfrost.org @DrFrostMaths Last modified: Contact the resource team: resources@drfrost.org @DrFrostResource Jasmine Vincent Divisibility Tests 7th January 2025
  • 2.
    How to usethese slides Dr Frost Learning is a registered charity in England and Wales (no 1194954) Slide Title Explanation Default Animations* Recap To be used as a prior knowledge check or to review prerequisite knowledge. Can be used as a starter or as part of the main lesson. Green click-to-reveal boxes. The Big Idea To be used to highlight key concepts or theorems. This could include the ‘why’ of the topic - including “real-life” contextual scenarios, or putting into context of other mathematical concepts (past and future). Usually in sequence with some green click-to-reveal boxes. Example To be modelled by the teacher. Solution animates in sequence. Test Your Understanding To be completed by students and used for Assessment for Learning, primarily using mini-whiteboards. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final answer to reveal full solution. Example Problem Pair To be used as ‘Example’ &‘Test Your Understanding’ above, within the same slide to provide scaffold via visible modelled solution. TYU column is blank initially, to focus attention on example. Reveal question by clicking ‘Test Your Understanding’ banner. Example animates in sequence. Click the header to reveal TYU question, then green click-to- reveal boxes. Quickfire Questions To be used as fluency practice. Multiple questions in rapid succession, for calculations that can be completed mentally. Often used for shorter questions/ formulae or to isolate a small part of the method. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final line to reveal full solution. Multi-choice Question To be used as a diagnostic question. Multiple choice questions, with plausible distractors, to allow teachers to diagnose misconceptions and errors in student thinking, then adapt their lesson accordingly. Arrow points to answer, on click. Exam Question To be completed by teacher or student. Green click-to-reveal boxes. Though many slides in this resource will have titles specific to the topic, the slide titles in the table below are used consistently within DFL resources for specific pedagogical purposes. Any atypical use of a slide type, including any change of animation* or intended use, will be outlined in the Teacher Notes for the slide. How to use these slides Dr Frost Learning is a registered charity in England and Wales (no 1194954) Slide Title Explanation Default Animations* Recap To be used as a prior knowledge check or to review prerequisite knowledge. Can be used as a starter or as part of the main lesson. Green click-to-reveal boxes. The Big Idea To be used to highlight key concepts or theorems. This could include the ‘why’ of the topic - including “real-life” contextual scenarios, or putting into context of other mathematical concepts (past and future). Usually in sequence with some green click-to-reveal boxes. Example To be modelled by the teacher. Solution animates in sequence. Test Your Understanding To be completed by students and used for Assessment for Learning, primarily using mini-whiteboards. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final answer to reveal full solution. Example Problem Pair To be used as ‘Example’ &‘Test Your Understanding’ above, within the same slide to provide scaffold via visible modelled solution. TYU column is blank initially, to focus attention on example. Example animates in sequence, followed by TYU question with green click-to-reveal boxes for solution steps. Quickfire Questions To be used as fluency practice. Multiple questions in rapid succession, for calculations that can be completed mentally. Often used for shorter questions/ formulae or to isolate a small part of the method. Green click-to-reveal boxes. For multi-step answers, reveal in parts or click final line to reveal full solution. Multi-choice Question To be used as a diagnostic question. Multiple choice questions, with plausible distractors, to allow teachers to diagnose misconceptions and errors in student thinking, then adapt their lesson accordingly. Arrow points to answer, on click. Exam Question To be completed by teacher or student. Green click-to-reveal boxes. Though many slides in this resource will have titles specific to the topic, the slide titles in the table below are used consistently within DFL resources for specific pedagogical purposes. Any atypical use of a slide type, including any change of animation* or intended use, will be outlined in the Teacher Notes for the slide.
  • 3.
    Teacher Notes Dr FrostLearning is a registered charity in England and Wales (no 1194954) Key Points Solution step – click to reveal Question/Discussion Prompt Key: ! To be written in books All slides include pedagogical detail in the ‘Notes’ section for each slide. Throughout the slides, this symbol refers to a web link. Unless otherwise specified, this will be to some functionality within DF. Future Links • Simplifying fractions • Identifying properties of a number based on it’s prime factorisation Prerequisite Knowledge • Identify multiples and factors • Find all factor pairs of a number • Identify common factors • Identify common multiples • Identify prime numbers • Prime factorisation
  • 4.
    Dr Frost Learningis a registered charity in England and Wales (no 1194954) Dr Frost Learning is a registered charity in England and Wales (no 1194954) Using the Dr Frost online platform Skills in this Lesson 163 Divisibility laws from 3 to 11 and dealing with larger divisors 163a Know if a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10. 163b Know if a number is divisible by 11. 163c Determine an unknown digit to make a number divisible by 3, 4, 6, 8 or 9. 163d Know if a number is divisible by a large number by combining divisibility rules. TEACHERS Generate a random worksheet involving skills in this PowerPoint (for printing or online task setting). STUDENTS Start an independent practice involving skills in this PowerPoint. drfrost.org/w/866 Clicking this box takes you to a single question practice for a subskill to allow you further Test Your Understanding opportunities. (e.g. drfrost.org/s/123a) drfrost.org/p/866 drfrost.org/s/123a
  • 5.
    Dr Frost Learningis a registered charity in England and Wales (no 1194954) Dr Frost Learning is a registered charity in England and Wales (no 1194954) Contents Prerequisite Check: Factors, Multiples and Primes Divisibility laws: and Divisibility laws: and Divisibility laws: and Divisibility laws: and Prime Numbers Missing digits Combining divisibility laws Exercise For lessons covering many concepts, please click the below to navigate quickly to the relevant part of the lesson.
  • 6.
    Prerequisite Check: Factors,Multiples and Primes 1 Write as a product of prime factors. 𝟏𝟖𝟎=𝟐𝟐 ×𝟑𝟐 ×𝟓 2 List the first multiples of 𝟖,𝟏𝟔,𝟐𝟒,𝟑𝟐,𝟒𝟎 3 List all the factors of 𝟏,𝟐,𝟑,𝟒,𝟔,𝟗,𝟏𝟐,𝟏𝟖,𝟑𝟔 4 5 Which of the numbers below are even? and 6 What is the smallest prime number? 𝟐 7 Is a prime or composite number? so is divisible by and making it a composite number 8 Identify the HCF of and . 𝟏𝟐 9 Identify the LCM of and . ? ? ? ? ? ? ? ? 10 Identify the HCF and LCM of and . HCF: LCM: ? Is a prime number? Explain why or why not. Yes, is a prime number because it has exactly factors; and itself. ? Show all solutions
  • 7.
    The Big Idea:Divisibility Rules Mrs Clark How do you know if a number is divisible by ? It will end in a 0, 2, 4, 6 or 8 It will be an even number Both correct! How do you know if a number is divisible by ? It will end in a … … or a ! Exactly! 163a drfrost.org/s/
  • 8.
    The Big Idea:Divisibility Rules Mrs Clark Do you know any other divisibility rules already? Factor Rule Last digit is even or Last digit is or Logan Any number that ends in a is divisible by Great! Let’s see if we can fill in the rest of the table Last digit is 163a drfrost.org/s/
  • 9.
    is divisible byand ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by and/or ? Yes, because it ends in an even digit. No, because it doesn’t end in a or a . Yes, because it ends in a . No, because it doesn’t end in a . Yes, because it ends in an even digit. ? ? ? ? ?
  • 10.
    The Big Idea:Divisibility Rules 163a drfrost.org/s/ Mrs Clark We can use divisibility laws for other numbers too! Some are more complicated than others… Nathan How am I meant to know if is divisible by something other than , or ? Factor Rule Last digit is even or Last digit is or Last digit is
  • 11.
    Do you noticeanything special about the highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Number Calculation Digit Sum Nathan They are all numbers in the times table; and their digit sum is also in the times table A number is divisible by if the sum of its digits if divisible by Is divisible by ? The digit sum of is: is divisible by therefore is divisible by
  • 12.
    Number Calculation DigitSum Do you notice anything special about these highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Logan I’ve noticed that the numbers in the times table have a digit sum of Will this always be true?
  • 13.
    The Big Idea:Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the times table in more detail Number Calculation Digit Sum Logan doesn’t fit the pattern; maybe the rule is the digits sum to either or ? Nathan You should test it out on a bigger number you know is divisible by – like Number Calculation Digit Sum is also in the times table; so the digit sum just needs to be divisible by as well! A number is divisible by if the sum of its digits if divisible by
  • 14.
    is not divisibleby or therefore is not divisible by or ? is divisible by therefore is divisible by . is divisible by therefore is divisible by is not divisible by therefore is not divisible by . is divisible by and therefore is divisible by both and . is divisible by but not by , therefore is divisible by but not by ? ? ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ? Is divisible by , or both? Is divisible by , or both? Is divisible by or both?
  • 15.
    The Big Idea:Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is 163a drfrost.org/s/
  • 16.
    Example: Checking divisibilityby or Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Is the number divisible by or ? The divisibly rules we’ve seen so far either use the last digit or the digit sum. The last digit of is . This means that is divisible by , but not or . The digit sum is: is divisible by and therefore is divisible by and . is divisible by and .
  • 17.
    Test Your Understanding FactorRule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Divisible by                  Tick the factor(s) that each number is divisible by ? ? ? ? ? ? ? Make a -digit number that matches each statement A multiple of Largest possible multiple of Multiple of and ? ? ? ? ? 163a drfrost.org/s/ 1 2 a b c Show all solutions
  • 18.
    The Big Idea:Divisibility Rules Mrs Clark All these numbers are divisible by ; do you notice anything special about them? 163a drfrost.org/s/ 216 10524 27340 99048 459136 It might help to look specifically at the last digits of each number Hannah If we treat the last digits as a -digit number; they are all multiples of A number is divisible by if the number formed by the last digits is divisible by Can you explain why this works?
  • 19.
    The Big Idea:Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the number 27340 27300 40 can be split into and is divisible by – we can tell this because it ends in two zeros Any number that ends in two zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by is divisible by therefore we know that is divisible by
  • 20.
    Example: Divisibility by163a drfrost.org/s/ 12345 12300 45 First; we split the into a multiple of and the rest of the number We know is divisible by because it is a multiple of is not divisible by therefore we know that is not divisible by Is the number divisible by ?
  • 21.
    is divisible by isalso divisible by Therefore is divisible by . is divisible by is not divisible by Therefore is not divisible by . is divisible by is divisible by Therefore is divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
  • 22.
    The Big Idea:Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The digit sum is a multiple of Last digit is 163a drfrost.org/s/
  • 23.
    The Big Idea:Divisibility Rules 163a drfrost.org/s/ True or False? All multiples of are even… True or False? All multiples of are even… True or False? All multiples of are even… False True True False False True Only some multiples of are even 3,6,9,12,15,18,21,24 ,27 ,30… Viktor The multiples of that are even are also multiples of ; and all the even numbers are multiples of A number is divisible by if it divisible by both and
  • 24.
    Example: Divisibility by Isthe number divisible by ? ends in an even number therefore it is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by both and therefore is divisible by A number is divisible by if it divisible by both and Is the number divisible by ? ends in an even number therefore it is divisible by The digit sum is: is not divisible by therefore is not divisible by is divisible by but not by therefore is not divisible by
  • 25.
    ends in aneven digit therefore is divisible by The digit sum is: is not divisible by therefore is not divisible by is not divisible by ends in an even digit therefore is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by ends in an odd digit therefore is not divisible by We don’t need to check if the number is divisible by as well is not divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
  • 26.
    The Big Idea:Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The digit sum is a multiple of Last digit is 163a drfrost.org/s/ Logan which is why this works… maybe there are other divisibility rules we can find that work in this way?
  • 27.
    The Big Idea:Divisibility Rules Mrs Clark Can you remember the divisibility rule for ? 163a drfrost.org/s/ A number is divisible by if the number formed by the last digits of the number is divisible by You can test if a number is divisible by in a similar way; this time we must look at the last digits to see if it is divisible by 23,416 Is divisible by ? If is divisible by then we will know the whole number is divisible by Viktor But how do we know if is divisible by ? To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer is divisible by , therefore is divisible by
  • 28.
    FYI: For YourInterest To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer For larger numbers we can use a similar divisibility rule for as we used for . 23 416 23000 416 can be split into and is divisible by – we can tell this because it ends in three zeros Any number that ends in three zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by is divisible by , we can tell by halving the number times (this works because Therefore is divisible by
  • 29.
    is divisible by isnot an integer therefore is not divisible by Therefore is not divisible by . is divisible by is an integer therefore is divisible by Therefore is divisible by . is divisible by is an integer therefore is divisible by Therefore is divisible by ? ? ? Quickfire Questions Is divisible by ? Is divisible by ? Is divisible by ?
  • 30.
    Test Your Understanding FactorRule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The last digits form a number divisible by The digit sum is a multiple of Last digit is 163a drfrost.org/s/ Divisible by                      Tick the factor(s) that each number is divisible by ? ? ? ? ? ? Are the statements below always, sometimes or never true? If a number is a multiple of it is also a multiple of If a number is a multiple of it is also a multiple of Always true Sometimes true ? ? 1 2 a b Show all solutions
  • 31.
    Example Test YourUnderstanding Mrs Clark The divisibility rules for and are slightly less intuitive A number is divisible by if when adding and subtracting its digits in an alternative pattern it makes a number divisible by . Is divisible by ? is divisible by therefore is divisible by Is divisible by ? drfrost.org/s/ 163b Is divisible by ? is divisible by (and any number except ) therefore is divisible by is not divisible by therefore is not divisible by ?
  • 32.
    Example Test YourUnderstanding Mrs Clark The divisibility rules for and are slightly less intuitive A number is divisible by if when you double the last digit and subtract it from the rest of the number, the result is divisible by . Is divisible by ? Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by Is divisible by ? Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by drfrost.org/s/ 163b ?
  • 33.
    Example Test YourUnderstanding Double the last digit: Number made by other digits: Subtract: Now we need to check Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by ? A number is divisible by if when you double the last digit and subtract it from the rest of the number, the result is divisible by . Is divisible by ? Double the last digit: Number made by other digits: Subtract: Now we need to check using the same process Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by Is divisible by ? drfrost.org/s/ 163b
  • 34.
    The Big Idea:Divisibility Rules Factor Rule Last digit is even or The digit sum is a multiple of The last 2 digits form a number divisible by 4 Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Here’s a summary of all the divisibility rules you have learned:
  • 35.
    Test Your Understanding FactorRule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by 7 The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Divisible by                    Tick the factor(s) that each number is divisible by ? ? ? ? ? ? 163a drfrost.org/s/ Show all solutions 163b
  • 36.
    Apart from theobvious instant checks (divisibility by , ), we usually only need to mentally check and to have a good ‘guess’ that a number is prime. Is it prime?  No Yes  No Yes  No Yes No! () N What is the largest prime factor we need to test before being certain a number is prime? We can use the square root of the number we are testing; for example so we would need to check up to 14. All composite numbers have a factor (other than 1) up to the square root. ? ? The Big Idea: Prime Numbers ? ? ? ? ? ? ? ? ? ? ? ? ? 163b drfrost.org/s/
  • 37.
    Example Test YourUnderstanding Find which possible digit(s) could go in the box to make  divisible by . Recall the divisibility law for A number is divisible by if it divisible by both and For the number to be divisible by it would need to end in a or For the number to divisible by the digits needs to add up to a multiple of The digit sum so far is… This is already a multiple of therefore the digit in the box also needs to be a multiple of The missing digit could be or Find which possible digit(s) could go in the box to make  divisible by . Recall the divisibility law for The two digit number  needs to be divisible by Consider the numbers in the times table around the number … … The missing digit could be or A number is divisible by if the number formed by the last digits of the number is divisible by drfrost.org/s/ 163c ?
  • 38.
    For the numberto be divisible by it needs to be divisible by and The number is already divisible by as it ends in a Digit sum: The digit sum is already a multiple of therefore the missing digit must also be divisible by : it could be or For the number to be divisible by the last digits need to make a multiple of The last digits could be: or Out of this list only some are multiples of : the missing digit could be or For the number to be divisible by the last digits needs to make a multiple of Thinking of the times table… …… The missing digit must be Test Your Understanding Find which possible digit(s) could go in the box to make divisible by . Find which possible digit(s) could go in the box to make divisible by . ? Find which possible digit(s) could go in the box to make divisible by . For the number to be divisible by the digit sum needs to be a multiple of Digit sum: To make the digit sum a multiple of it would need to sum to : the missing digit must be Find which possible digit(s) could go in the box to make divisible by . ? ? ? 163c drfrost.org/s/ 1 2 3 4
  • 39.
    If we wantto check if a number is divisible by , we can show it is divisible by and , they are coprime and have a product of Are these statements true or false? If we want to show that a number is divisible by , we can show it is divisible by and . If we want to show that a number is divisible by , we can show it is divisible by and . The Big Idea: Combining Divisibility Rules False True True False 24 48 7 2 9 6 120 This is sometimes true; all of the multiples of are divisible by and . However, there are common multiples of and that are not divisible by . E.g. is divisible by and but not by . 36 We need to pick two numbers which are coprime, i.e. do not share any factors. Therefore, how should we test if a number is divisible by ? ?
  • 40.
    Quickfire Questions What divisibilityrules would we use to test divisibility by: and and and and and and and and 163d drfrost.org/s/ and and
  • 41.
    Example Test YourUnderstanding Find which possible digit(s) could go in the box to make divisible by . For the number to be divisible by it must be divisible by and Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or Divisible by Last digits need to be a multiple of : multiples of : , , , , Digits that fit both rules and can therefore be the missing digit: or Find which possible digit(s) could go in the box to make divisible by For the number to be divisible by it must be divisible by and Divisible by Must end in a or Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or The only digit that fits both rules and is therefore the missing digit: drfrost.org/s/ 163d ? ? ? ?
  • 42.
    Exercise (Available asa separate worksheet) Show all solutions 1 Tick the factor(s) that each number is divisible by Divisible by Prime                                     ? ? ? ? ? ? [JMC 2011 Q2] How many of the integers are multiples of ? [ ] [ ] [ ] [ ] [ ] Answer: (all numbers have a digit sum that is a multiple of ) 2 [JMC 2004 Q2] Which of the following numbers is exactly divisible by ? 3 Answer: ? ? ?
  • 43.
    The number needsto be divisible by and ; the digit sum is and needs to be a multiple of Answer: [JMC 2016 Q11] Which of the following statements is false? [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [ ] is a multiple of [JMC 2003 Q13] was a prime year, since is a prime number. In the following ten years there was just one prime year. Which was it? Hint: Use your divisibility rules! [ ] [ ] [ ] [ ] [ ] Answer: is divisible by is divisible by and is divisible by is divisible by [JMC 1999 Q17] The -digit number is a multiple of . Which digit is represented by ? Answer: Therefore which creates a final sum of Exercise (Available as a separate worksheet) Show all solutions 4 5 6 Answer: ( is not a multiple of ) 7 8 ? ? Find which possible digit(s) could go in the box to make divisible by . Current digit sum is ; this is already a multiple of Answer: or ? Find which possible digit(s) could go in the box to make divisible by ? ?
  • 44.
    [JMC 2000 Q17] Thefirst and third digits of the five- digit number are the same. If the number is exactly divisible by , what is the sum of its five digits? Answer: Digit sum: must be an odd number between and , and the only odd multiple of in this interval is . [JMC 1997 Q20] A four-digit number was written on a piece of paper. The last two digits were then blotted out (as shown). If the complete number is exactly divisible by three, by four, and by five, what is the sum of the two missing digits? Answer: The missing digits are and [JMC 2012 Q23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number . How many of the numbers on Peter’s list are prime? [ ] [ ] [ ] [ ] [ ] Answer: [Pink Kangaroo 2020 Q16] The digits from to are randomly arranged to make a -digit number. What is the probability that the resulting number is divisible by ? Answer: Exercise (Available as a separate worksheet) Show all solutions 9 ? ? ? ? 10 11 12 8 6
  • 45.
    [SMC 2012 Q6] Whatis the sum of the digits of the largest -digit palindromic number which is divisible by ? Palindromic numbers read the same backwards and forwards, e.g. . Answer: For the number to be divisible by it eithers ends in a or a . It cannot end in a as the first digit cannot be . Therefore, the number is 5**5 The digit sum needs to be a multiple of , this is true if * or therefore the largest is when * which makes the digit sum Exercise (Available as a separate worksheet) Show all solutions 13 The letters , and stand for non-zero digits. The integer ‘’ is a multiple of ; the integer ‘’ is a multiple of ; and the integer ‘’ is a multiple of . What is the integer ‘’? Answer: • is a multiple of , therefore is a multiple of • is a multiple of , the digit sum for would be the same therefore is a multiple of and • Hence is a multiple of • is a multiple of ; it must be a multiple of and as the digits are non-zero and is a multiple of • The digit multiples of that start with a () are: and • Only has a as a multiple of • Therefore and N ? ?

Editor's Notes

  • #6 Teacher notes: Prerequisite check
  • #7 Teacher notes: Get students to recall any divisibility tests they already know
  • #8 Teacher notes: We will add to this table as we go through this section of the lesson. A blank printable version is available on slide 10 and a filled in printable version is available on slide 38
  • #9 Teacher notes: Blank version to print – a filled in version is also available to print on slide XXXX
  • #10 Teacher notes: Quickfire questions to check understanding of divisibility laws for 2, 5 and 10
  • #11 Teacher notes: Summary so far
  • #12 Teacher notes: Introduction to the divisibility law for 3
  • #13 Teacher notes: Linking to divisibility law for 9
  • #14 Teacher notes: Looking at the divisibility law for 9 in more depth
  • #15 Teacher notes: Quickfire questions to check understanding of divisibility laws for 3 and 9
  • #16 Teacher notes: Summary so far
  • #17 Teacher notes: Example to show how to check multiple divisibility laws
  • #18 Teacher notes: Questions to test students understanding so far. There is a printable version of the first question available on the next slide.
  • #19 Teacher notes: Printable version of question 1 on previous slide
  • #20 Teacher notes: Introduction to divisibility law for 4. Explanation on next slide.
  • #21 Teacher notes: Explanation of why the divisibility law for 4 works.
  • #22 Teacher notes: Further example to show how to use the divisibility law for 4
  • #23 Teacher notes: Quickfire questions to check understanding of divisibility law for 4
  • #24 Teacher notes: Summary so far
  • #25 Teacher notes: Introduction to the divisibility law for 6
  • #26 Teacher notes: Further examples for the divisibility law for 6
  • #27 Teacher notes: Quickfire questions to check understanding of the divisibility law for 6
  • #28 Teacher notes: Summary so far and linking the divisibility law for 6 to future laws
  • #29 Teacher notes: Introduction to divisibility law for 8
  • #30 Teacher notes: Explanation of the divisibility law for 8 and how it links to the divisibility law for 4
  • #31 Teacher notes: Quickfire questions to check understanding of the divisibility law for 8
  • #32 Teacher notes: Test your understanding questions to check understanding of the laws covered so far
  • #33 Teacher notes: Introduction to the divisibility law for 11 This divisibility test was discovered through mathematical exploration and an understanding of modular arithmetic
  • #34 Teacher notes: Introduction to the divisibility law for 7 This divisibility test was discovered through mathematical exploration and an understanding of modular arithmetic
  • #35 Teacher notes: Further example using the divisibility law for 7 where the process has to be repeated This divisibility test was discovered through mathematical exploration and an understanding of modular arithmetic
  • #36 Teacher notes: Completed printable version of this table available on the next slide
  • #37 Teacher notes: To print
  • #38 Teacher notes: Test your understanding questions for all the divisibility laws covered
  • #39 Teacher notes: Using divisibility laws to check if a number is prime
  • #40 Teacher notes: Example problem pair showing how to find missing digits
  • #41 Teacher notes: Test your understanding questions for missing digit questions
  • #42 Teacher notes: Explanation of how to combine divisibility laws to check divisibility by larger numbers
  • #43 Teacher notes: Quickfire questions based on the previous slide
  • #44 Teacher notes: Example problem pair for missing digit numbers where the factor is a number that requires combining divisibility laws
  • #45 Teacher notes: Click show all solutions to reveal all the answers
  • #46 Teacher notes: Click show all solutions to reveal all the answers
  • #47 Teacher notes: Click show all solutions to reveal all the answers
  • #48 Teacher notes: Click show all solutions to reveal all the answers