**Common Factors and Common Multiples**

**Common Factors**. *When we find the factors of two or more numbers, we find some factors are same for the numbers. They are the common factors.*

Example. Find the common factors of 4, 12 and 16.

Solution. Factors of 4 are 1, 2 and 4.Factors of 12 are 1, 2, 3, 4, 6 and 12.

Factors of 16 are 1, 2, 4, 8 and 16.

Clearly, 1, 2 and 4 are the common factors of 4, 12, and 16.

Example. Find the common factors of 14, 25 and 76.

Solution. Factors of 14 are 1, 2, 7 and 14.Factors of 25 are 1, 5 and 25.

Factors of 76 are 1, 2, 4, 19 and 76.

Clearly, these three numbers have only 1 as the common factor.

Example. Find the common factors of 75, 60 and 210.

Solution. Factors of 75 are 1, 3, 5, 15, 25 and 75.Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60.

Factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Thus, common factors of 75, 60 and 210 are 1, 3, 5 and 15.

**Common Multiples**. *The multiples that are common to two or more numbers are called the common multiples of those numbers.*

Example. What are the common multiples of 4 and 6?

Solution. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …The multiples of 6 are 6, 12, 18, 24, 30, 36, …

Common multiples of 4 and 6 are 12, 24, 36, …

Example. Find the common multiples of 3, 5 and 6.

Solution. Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, …

Multiples of 6 are 6, 12, 18, 24, 30, …

Common multiples of 3, 5 and 6 are 30, 60, …

Example. Find the common multiples of 3, 4 and 9.

Solution. Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, ….Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, …

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, …

Common multiples of 3, 4 and 9 are 36, 72, 108, …

**Perfect Numbers**. *A number for which sum of all its factors is equal to twice the number is called a perfect number*

Example. Show that the 6 and 28 are perfect numbers.Solution. The factors of 6 are 1, 2, 3 and 6. Also,1 + 2 + 3 + 6 = 12 = 2 × 6.

All the factors of 28 are 1, 2, 4, 7, 14 and 28. Adding these we have,

1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.

Hence, the numbers 6 and 28 are perfect numbers.

**Exercise 1**. Find the common factors of (a) 20 and 28 (b) 15 and 25 (c) 35 and 50 (d) 56 and 120

**Exercise 2**. Find the common factors of (a) 4, 8 and 12 (b) 5, 15 and 25

**Exercise 3**. Find first three common multiples of (a) 6 and 8 (b) 12 and 18

**Exercise 4**. Write all the numbers less than 100 which are common multiples of 3 and 4.

**Exercise 5**. Which of the following numbers are co-prime?

(a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 68 (e) 216 and 215 (f) 81 and 16

**Exercise 6**. A number is divisible by both 5 and 12. By which other number will that number be

always divisible?

**Exercise 7**. A number is divisible by 12. By what other numbers will that number be divisible?