Factors and Multiples
Divisibility Rules
The divisibility rules are the shorthand rules to identify if the given integers are divisible by the divisor.
List of divisibility rules
Divisible by 2:
 A number is said to be divisible by 2 if the last digits of the number is 0, 2, 4, 6 and 8.

Examples:24, 380, 4502 are divisible by 2.
Divisible by 3:
 A number is said to be divisible by 3 if the sum of the digits is divisible by 3

Examples: Consider the number 342. The sum of the digits is 3 + 4 + 2 = 9 Since the number 9 is divisible by 3. Therefore, 342 is also divisible by 3.
Divisible by 4:
 A number is said to be divisible by 4 if the last two digits of the number is divisible by 4.

Examples:Consider the number 420.
Divisible by 5:
 A number is said to be divisible by 5 if the last digits of the number is either 0 or 5.

Examples:25, 340, 765 are divisible by 5.
Divisible by 6:
 A number is said to be divisible by 6 if it is divisible by both 2 and 3.

Examples:Consider the number 84.
Divisible by 9:
 A number is said to be divisible by 9 if the sum of the digits is divisible by 9.

Examples:Consider the number 342.
Divisible by 10:
 A number is said to be divisible by 10 if the last digits of the number is 0.

Examples:240, 380, 450 are divisible by 10.
The last two digits of the number is 20.
Since the number 20 is divisible by 4. Therefore, 420 is also divisible by 4.
The last digit is 4, 84 is divisible by 2.
The sum of the digit is 8 + 4 = 12. The number 12 is divisible by 3. Therefore, 84 is divisible by 3.
The number 84 is divisible by both 2 and 3. Therefore, 84 is also divisible by 6.
The sum of the digits is 3 + 4 + 2 = 9
Since the number 9 is divisible by 9. Therefore, 342 is also divisible by 9.
Examples:
Prime Factorization
 The process of breaking down the number as the product of prime numbers.
 The two methods of prime factorization are:
 Factor Tree
 Division Method
Factor Tree
Factor tree is a tree diagram which is used to find the prime factors of a number
Steps to draw a factor tree:
 Each branch of the tree is split into two factors.
 It stops when the factors at the end of the branch is a prime number.
Examples:
 In the above example, it is observed that:
 The product of 6 and 18 is equal to 108. Therefore, the 6 and 18 are the factors of 108.
 The numbers 2 and 3 are the prime numbers. So, the factor tree stops.
 Therefore, 108 = 2 Ă— 2 Ă— 3 Ă— 3 Ă— 3.
Division Method
It is a process of finding the prime factors of a number by dividing the number.
Steps for the division method of prime factorization:
 Divide the number by the smallest prime numbers.
 Repeat the process till we get 1 as a factor.
Examples:
 In the above example the number 64 is first divided by the smallest prime number which is 2.
 Then 32 is divided by the smallest prime number 2 to get 16.
 The process of division is continued unit we get 1 as a factor.
 Therefore, 64 = 2 Ă— 2 Ă— 2 Ă— 2 Ă— 2 Ă— 2
Commom Mistakes:
While making a factor tree the most common mistake is that the numbers are split in such a way that their sum is equal to the number.
Examples: