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!
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
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.
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.
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
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