Can I Divide This Number By That Number

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Mathematical divisibility rules;
correlates with Glencoe Mathematics Course 1: 1-2 and Pre-Algebra: 4-1

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Can I Divide This Number By That Number

  1. 1. Can I divide this number by that number (without leaving a remainder) ?
  2. 2. There’s an easy way to tell! Use divisibility rules. <ul><li>DIVISIBILITY means </li></ul><ul><li>one whole number (…-3, -2, -1, 0, +1,+2, +3…) </li></ul><ul><li>can be divided by another whole number </li></ul><ul><li>without leaving a remainder or </li></ul><ul><li>(to put it another way) </li></ul><ul><li>leaving a remainder of 0 . </li></ul>
  3. 3. Divisibility Rules is always undefined . A number divided by 0 it ends in 0. 10 the sum of its digits is divisible by 9. 9 its last 3 digits (taken together as a three-digit number) are divisible by 8. 8 the original number without the last digit minus 2x last digit, repeated until the difference is 20 or less provides an answer (difference) divisible by 7. 7 it is even and the sum of its digits is divisible by 3. 6 it ends in 0 or 5. 5 its last two digits (viewed as a two-digit number) are divisible by 4. 4 the sum of its digits is divisible by 3. 3 its last digit is even: 0, 2, 4, 6, or 8. 2 all of the time. 1 If… A number can be divided by….
  4. 4. Can I divide this number by 1? <ul><li>YES! </li></ul><ul><li>EVERY NUMBER can be divided by 1. </li></ul><ul><li>In fact, the number 1 is quite magical , because when you divide any number by 1, the quotient (answer) is ALWAYS the number you started with. </li></ul><ul><li>328 ÷ 1 = 328 </li></ul><ul><li>This works in multiplication too. </li></ul><ul><li>328 x 1 = 328 </li></ul><ul><li>And this special number magic has a name. The Multiplicative Identity Property. (We’ll learn more about that later!) </li></ul>
  5. 5. Can I divide this number by 2? <ul><li>YES </li></ul><ul><li>If the last digit in the number is even. </li></ul><ul><li>Look at the last digit in the number. </li></ul><ul><li>If it is a 0, 2, 4, 6, or 8 </li></ul><ul><li>the number is even and you can divide it by 2! </li></ul><ul><li>No </li></ul><ul><li>If the last digit in the number is odd. </li></ul><ul><li>Look at the last digit in the number. </li></ul><ul><li>If it is a 1, 3, 5, 7, or 9 </li></ul><ul><li>The number is odd and you can NOT divide it by 2! </li></ul>
  6. 6. Is 328 divisible by 2? <ul><li>What about 328? </li></ul><ul><li>Use the rule. </li></ul><ul><li>The last digit in 328 is 8. </li></ul><ul><li>8 is an even number . </li></ul><ul><li>SO, I know that </li></ul><ul><li>328 is divisible by 2. </li></ul><ul><li>328 ÷ 2 = 164 </li></ul>
  7. 7. Is 327 divisible by 2? <ul><li>Use the rule. </li></ul><ul><li>The last digit in 327 is 7. </li></ul><ul><li>7 is an odd number . </li></ul><ul><li>SO, I know that </li></ul><ul><li>327 is NOT divisible by 2. </li></ul><ul><li>327 ÷ 2 = 163 R1 </li></ul>
  8. 8. Can I divide this number by 3? <ul><li>YES </li></ul><ul><li>If the sum of the digits in the number is divisible by 3. </li></ul>
  9. 9. Is 327 divisible by 3? <ul><li>Use the rule. </li></ul><ul><li>Add the digits of 327 together. </li></ul><ul><li>3 + 2 + 7 = 12 </li></ul><ul><li>Divide the sum of the digits by 3. </li></ul><ul><li>12 ÷ 3 = 4 Remainder 0 </li></ul><ul><li>There is no remainder. </li></ul><ul><li>SO, I know that </li></ul><ul><li>327 IS divisible by 3! </li></ul><ul><li>See. It is. </li></ul><ul><li>109 R 0 </li></ul><ul><li>3) 327 3 + 2 + 7 = 12 </li></ul><ul><li>- 3 12 ÷ 3 = 4 R 0 </li></ul><ul><li>02 </li></ul><ul><li>- 0 </li></ul><ul><li>27 </li></ul><ul><li>- 27 </li></ul><ul><li>0 </li></ul>
  10. 10. Is 329 divisible by 3? <ul><li>Use the rule. </li></ul><ul><li>Add the digits of 329 together. </li></ul><ul><li>3 + 2 + 9 = 14 </li></ul><ul><li>Divide the sum of the digits by 3. </li></ul><ul><li>14 ÷ 3 = 4 Remainder 2 </li></ul><ul><li>There is a remainder… </li></ul><ul><li>SO, I know that </li></ul><ul><li>329 IS NOT divisible by 3! </li></ul><ul><li>See. It isn’t. </li></ul><ul><li>109 R 2 </li></ul><ul><li>3) 329 3 + 2 + 7 = 12 </li></ul><ul><li>- 3 12 ÷ 3 = 4 R 0 </li></ul><ul><li>02 </li></ul><ul><li>- 0 </li></ul><ul><li>29 </li></ul><ul><li>- 27 </li></ul><ul><li>2 </li></ul>
  11. 11. Can I divide this number by 4? <ul><li>YES </li></ul><ul><li>If the number formed by the last two digits is divisible by 4. </li></ul><ul><li>(YIKES! What does that mean?) </li></ul><ul><li>Ignore all the digits in the number, but the last two. </li></ul><ul><li>Look at the last two digits as a number. </li></ul><ul><li>Can you divide it by 4? </li></ul><ul><li>Good. Then you can divide the entire number by 4. </li></ul>
  12. 12. Is 327 or 328 divisible by 4? <ul><li>Use the rule. </li></ul><ul><li>The last two digits in 328 are 28. </li></ul><ul><li>28 is divisible by 4 . </li></ul><ul><li>28 ÷ 4 =7 Remainder 0 </li></ul><ul><li>SO, I know that </li></ul><ul><li>328 is divisible by 4 . </li></ul><ul><li>328 ÷ 4 = 82 </li></ul><ul><li>Use the rule. </li></ul><ul><li>The last two digits in 327 are 27. </li></ul><ul><li>27 is NOT divisible by 4 . </li></ul><ul><li>27 ÷ 4 = 6 Remainder 3 </li></ul><ul><li>SO, I know that </li></ul><ul><li>327 is NOT divisible by 4. </li></ul><ul><li>327 ÷ 4 = 81 R3 </li></ul>
  13. 13. Can I divide this number by 5? <ul><li>YES </li></ul><ul><li>If the digit in the ones place (the last digit to the right) is a 0 or a 5. </li></ul>
  14. 14. Is 325 or 330 or 328 divisible by 5? <ul><li>Use the rule. </li></ul><ul><li>The last digit in 325 is 5 . </li></ul><ul><li>So, I know that </li></ul><ul><li>325 is divisible by 5. </li></ul><ul><li>325 ÷ 5 = 65 </li></ul><ul><li>and </li></ul><ul><li>Use the rule. </li></ul><ul><li>The last digit in 330 is 0 . </li></ul><ul><li>So, I know that </li></ul><ul><li>330 is divisible by 5. </li></ul><ul><li>330 ÷ 5 = 66 </li></ul><ul><li>Use the rule. </li></ul><ul><li>The last digit in 328 is not 0. </li></ul><ul><li>The last digit in 328 is not 5. </li></ul><ul><li>The last digit in 328 is 8. </li></ul><ul><li>So, I know that </li></ul><ul><li>328 is NOT divisible by 5. </li></ul><ul><li>328 ÷ 5 = 65 R3 </li></ul>
  15. 15. Can I divide this number by 6? <ul><li>YES </li></ul><ul><li>If the number is divisible by both 2 and 3. </li></ul><ul><li>NO </li></ul><ul><li>If the number is divisible by 2, but NOT by 3. </li></ul><ul><li>If the number is divisible by 3, but NOT by 2. </li></ul>
  16. 16. Is 328 divisible by 6? <ul><li>Is 328 divisible by 2? </li></ul><ul><li>Use the rule. </li></ul><ul><li>The digit in the ones place is 8. </li></ul><ul><li>8 is an even number. </li></ul><ul><li>SO, I know that </li></ul><ul><li>328 is divisible by 2. </li></ul><ul><li>328 ÷ 2 = 164 </li></ul><ul><li>Is 328 divisible by 3? </li></ul><ul><li>Use the rule. </li></ul><ul><li>Add the digits of 328 together. </li></ul><ul><li>3 + 2 + 8 = 13. </li></ul><ul><li>Divide the sum of the digits by 3. </li></ul><ul><li>13 ÷ 3 = 4 R1 </li></ul><ul><li>So, I know that </li></ul><ul><li>328 is NOT divisible by 3. </li></ul>So, because 328 is NOT divisible by both 2 and 3, I know that 328 is NOT divisible by 6 .
  17. 17. Can I divide this number by 7? <ul><li>YES </li></ul><ul><li>BUT, this is tricky. </li></ul><ul><li>If </li></ul><ul><li>The number without the last digit attached </li></ul><ul><li>minus the last number times 2 </li></ul><ul><li>is less than 20 and divisible by 7 the whole number is divisible by 7. </li></ul><ul><li>If the number without the last digit attached </li></ul><ul><li>minus the last number times 2 </li></ul><ul><li>is more than 20, then do the same thing again over and over again until the difference is 20 or less. </li></ul><ul><li>Take the new number (the difference) without the last digit attached, </li></ul><ul><li>subtract the last number times 2 </li></ul><ul><li>If the difference is less than 20, and the number is divisible by 7, then the original number is divisible by 7. </li></ul>
  18. 18. Is 329 divisible by 7? <ul><li>Use the rule. </li></ul><ul><li>Take the number without the last digit attached. </li></ul><ul><li>32 </li></ul><ul><li>Subtract the last digit 9 times 2. </li></ul><ul><li>9 x 2 = 18 </li></ul><ul><li>32 – 18 = 14 (difference) </li></ul><ul><li>If the difference is less than 20, and it is divisible by 7, the original number is divisible by 7 . </li></ul><ul><li>14 is less than 20, 14 is divisible by 7. </li></ul><ul><li>14 ÷ 7 = 2 </li></ul><ul><li>So, I know that </li></ul><ul><li>329 is divisible by 7. </li></ul><ul><li>329 ÷ 7 = 47 </li></ul>
  19. 19. Is 328 divisible by 7? <ul><li>Use the rule. </li></ul><ul><li>Take the number without the last digit attached. </li></ul><ul><li>32 </li></ul><ul><li>Subtract the last digit 8 x2. </li></ul><ul><li>8 x 2 = 16 </li></ul><ul><li>32 – 16 = 16 (difference) </li></ul><ul><li>If the difference is less than 20 is it divisible by 7, the original number is divisible by 7. </li></ul><ul><li>16 is less than 20; 16 is NOT divisible by 7. </li></ul><ul><li>SO, I know that </li></ul><ul><li>328 is not divisible by 7. </li></ul>
  20. 20. Can I divide this number by 8? <ul><li>YES </li></ul><ul><li>If it’s last three digits taken together as a number are divisible by 8. </li></ul>
  21. 21. Is 4,328 divisible by 8? <ul><li>Use the rule. </li></ul><ul><li>Take the last 3 digits. </li></ul><ul><li>328 </li></ul><ul><li>See if they are divisible by 8 without a remainder . </li></ul><ul><li>If it is, the whole number is divisible by 8. </li></ul><ul><li>328 is divisible by 8. </li></ul><ul><li>So, I know that </li></ul><ul><li>4,328 is divisible by 8 . </li></ul><ul><li>4,328 ÷ 8 = 541 </li></ul><ul><li>See, it works. </li></ul><ul><li>41 541 </li></ul><ul><li>8)328 8)4328 </li></ul><ul><li>- 32 - 40 </li></ul><ul><li>08 32 </li></ul><ul><li>- 8 - 32 </li></ul><ul><li>0 08 </li></ul><ul><li>- 8 </li></ul><ul><li>0 </li></ul>
  22. 22. Can I divide this number by 9? <ul><li>YES </li></ul><ul><li>If the sum of the digits is divisible by 9. </li></ul>
  23. 23. Is 324 or 327 divisible by 9? <ul><li>Use the rule. </li></ul><ul><li>Add the digits in the number 324. </li></ul><ul><li>3 + 2 + 4 = 9 </li></ul><ul><li>Divide the sum of the digits by 9. </li></ul><ul><li>9 ÷ 9 = 1 R0 </li></ul><ul><li>If the remainder is 0, the number is divisible by 9. </li></ul><ul><li>So I know that, </li></ul><ul><li>324 is divisible by 9. </li></ul><ul><li>324 ÷ 9 = 36 </li></ul><ul><li>Use the rule. </li></ul><ul><li>Add the digits in the number 327. </li></ul><ul><li>3 + 2 + 7 = 12 </li></ul><ul><li>Divide the sum of the digits by 9. </li></ul><ul><li>12 ÷ 9 = 1 R3 </li></ul><ul><li>If the remainder is 0, the number is divisible by 9. </li></ul><ul><li>So I know that, </li></ul><ul><li>327 is not divisible by 9. </li></ul><ul><li>327 ÷ 9 = 36 R3 </li></ul>
  24. 24. Can I divide this number by 10? <ul><li>YES </li></ul><ul><li>If the last digit in the number is a 0. </li></ul>
  25. 25. Is 328 divisible by 10? <ul><li>Use the rule. </li></ul><ul><li>The last digit of 328 is 8. </li></ul><ul><li>8 is not 0 (zero). </li></ul><ul><li>So, I know that </li></ul><ul><li>328 is NOT divisible by 10. </li></ul><ul><li>But, </li></ul><ul><li>The last digit in 320 is 0 . </li></ul><ul><li>So, I know that </li></ul><ul><li>320 is divisible by 10. </li></ul><ul><li>And </li></ul><ul><li>The last digit in 330 is 0. </li></ul><ul><li>So, I know that </li></ul><ul><li>330 is divisible by 10. </li></ul>
  26. 26. What happens when I divide a number by 0? <ul><li>Remember fact families. </li></ul><ul><li>Division </li></ul><ul><li>12 ÷ 4 = 3 or 12 = 3 </li></ul><ul><li> 4 </li></ul><ul><li>Multiplication </li></ul><ul><li>is the inverse (opposite). </li></ul><ul><li>quotient x divisor = dividend </li></ul><ul><li>x 4 = 12 </li></ul><ul><li>or, working backwards, </li></ul><ul><li>12 = 4 x 3 </li></ul><ul><li>BUT, when I try to divide with 0. </li></ul><ul><li>12 ÷ 0 or 12 = undefined not 0 </li></ul><ul><li>0 quotient </li></ul><ul><li>because, when I multiply (do the opposite or inverse). </li></ul><ul><li>no quotient x 0 will = 12 and </li></ul><ul><li>0 x 0 = 0 not 12 </li></ul><ul><li>There is no number in the whole world I can put in the quotient’s place as a factor that will equal 12, when I multiply it by 0. The product will always be 0. </li></ul><ul><li>So, when I divide a number by 0, </li></ul><ul><li>the result is called “undefined.” </li></ul>
  27. 27. Congratulations! <ul><li>Now you know the short cut for determining if a number can be divided by another number! </li></ul><ul><li>This will come in handy when you are finding prime and composite numbers, prime factors (prime factorization), greatest common factors (GCF), least common multiples (LCM), and need to find equivalent fractions. </li></ul>
  28. 28. Notes for teachers on texts correlation and design: <ul><li>Correlates with Glencoe Mathematics (Florida Edition) texts: </li></ul><ul><li>Mathematics: Applications and Concepts Course 1: (red book) </li></ul><ul><li>Chapter 1 Lesson 2 Divisibility Patterns </li></ul><ul><li>Mathematics: Applications and Concepts Course 2: (blue book) </li></ul><ul><li>Prerequisite skills </li></ul><ul><li>Pre-Algebra: (green book) </li></ul><ul><li>Chapter 4 Lesson 1: Factors and Monomials </li></ul><ul><li>This slide presentation was created using Microsoft Office PowerPoint 2003 part of Microsoft Office Standard Version for Students and Teachers </li></ul><ul><li>Clip Art came from Microsoft Office PowerPoint 2003 without exception. </li></ul><ul><li>Thank you for viewing this slide presentation. Thanks to my colleague, Sarah, for sharing much of this information with me. I hope you will find it of help to your students. Taleese </li></ul><ul><li>For more information on my math class see http://www.walsh.edublogs.org </li></ul>

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