This document discusses different types of numbers and arithmetic concepts. It covers: - Types of numbers including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. - Properties of even and odd numbers as well as positives and negatives under addition, subtraction, and multiplication. - Divisibility rules for numbers being divisible by 1 through 10 as well as squares. - Strategies for solving word problems involving remainders, including picking numbers, back-solving, and elimination.

1. Arithmetic
All about the art & science of numbers

2. Number systems
 What is the difference between number sense and counting?
(Brahmagupta, 596 AD)

3. A history of number systems
46,206 =
One important part of every number system is its base.

4. Types of numbers
What types of numbers can you recall?

5. Types of numbers
Term
Definition
Examples
Natural numbers
Counting numbers.
1, 2, 3, 4, 5, 6
Whole numbers
Counting numbers and zero.
0, 1, 2, 3
Integers
Whole numbers, their
opposites and zero.
–2, –1, 0, 1, 2, 3
Rational Numbers
Repeating or terminal
decimals.
1/3, 1/2
Irrational numbers
Non-repeating and non
terminating decimals
22/7 ,
Real numbers
Every number on the
number line
Anything!

6. The set of numbers

7. Consecutive numbers and the number
line

8. Odd and Even Numbers
Addition
Subtraction
even + even = even even – even = even
Multiplication
even . even = even
odd + odd = even
odd – odd = even
odd . odd = odd
even + odd = odd
even – odd = odd
even . odd = even
odd +even = odd
odd – even = odd
odd . even = even

9. Positives and Negatives
Multiplication
Division
positive . positive = positive
positive / positive = positive
negative . negative = positive negative / negative = positive
positive . negative = negative positive / negative = negative
negative . positive = negative negative / positive = negative

10. Divisibility and Remainders
What rules do you remember for divisibility?
 For example all even numbers are divisible by 2.

11. Divisibility and Remainders
 All whole numbers are divisible by 1.
 A number that ends in an even digit is divisible by 2.
 A number is divisible by 3 if its digits add up to a number divisible by 3.
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For example, 384 is divisible by 3 because 3 + 8 + 4 = 15, and 15 is
divisible by 3.
A number is divisible by 4 if its last two digits are divisible by 4. The
number 5,764 is divisible by 4 because 64 is divisible by 4.
A number is divisible by 5 if it ends in 0 or 5.
A number is divisible by 6 if it is even and divisible by 3. This rule is a
combo of rules 2 and 3.
Sadly, there is no rule for 7.
A number is divisible by 8 if its last three digits are divisible by 8. For
example, 1,249,216 is divisible by 8 because 216 is divisible by 8.
A number is divisible by 9 if its digits add up to a number divisible by 9.
The number 2,952 is divisible by 9 because 2 + 9 + 5 + 2 = 18.
A number is divisible by 10 if it ends in 0.

12. Divisibility and Remainders: Squares

13. Divisibility and Remainders: Squares
0 -> then the ending digit is a 0
1 -> then the ending digit is 1 or 9.
4 -> then the ending digit is 2 or 8.
5 -> then the ending digit is a 5.
6 -> then the ending digit is 4 or 6.
9 -> then the ending digit is 3 or 7.
C. After finding the last digit (or possibility between two digits) mentally chop off the last
two digits and focus on the remaining digits.

14. Example 1:
 Q) The number n is a 2 digit integer. When n is divided by 5
it leaves a remainder of 4 and when n is divided by 9, it leaves
a remainder of 7. What is the value of n?

15. Strategies
1a) Picking Numbers
Step 1. Pick Simple Numbers and substitute for variables.
Step 2. Try Them Out
 Try out all the answer choices using the numbers you picked,
eliminating those that gave you a different result.
Step 3. Try Different Values
 If more than one answer choice works, use different values
and start again
 1b) Recognition by prolonged and multiple use of Picking
Numbers

16. 1a) Picking Numbers

17. 1b) Recognition
Use different and easy Number Picks to eliminate any
possible integer answers.

18. 2) Back-solving
Step 1: Estimate the answer.
Step 2: Plug in an answer.
Step 3: Keep plugging in till you find one that works.
Step 4: Try out all answers.
Q. If x is an integer and 2 is the remainder when 3x +
4 is divided by 5, then x could equal
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

19. 3) Elimination