**Factors** and Multiples

**Factors**. *A factor of a number is an exact divisor of that number*.

Following facts are important:

- A factor of a number is an exact divisor of that number.
- Every number is a factor of itself.
- 1 is a factor of every number.
- Number of factors of a given number are finite.
- Every factor of a number is less than or equal to the given number.
- If a number is divisible by another number, then it is divisible by each of the factors of that number.

**Multiples**. *It is a number formed by multiplying that number by any whole numbers*.

Following facts are immediate:

- Every multiple of a given number is greater than or equal to that number.
- Every number is a multiple of itself.
- Number of multiples of a given number is infinite.
- Every number is a multiple of itself.

Example. Write all the factors of 68.

Solution. We note that68 ÷ 1 = 68

68 ÷ 2 = 34

68 ÷ 4 = 17

Thus, all the factors of 68 are 1, 2, 4, 17, 34 and 68.

Example. Find the factors of 36.

Solution. We note that36 ÷ 1 = 36

36 ÷ 2 = 18

36 ÷ 3 = 12

36 ÷ 4 = 9

36 ÷ 6 = 6

Thus, the factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Example. Write first five multiples of 6.

Solution. The required multiples are:6 × 1 = 6

6 × 2 = 12

6 × 3 = 18

6 × 4 = 24

6 × 5 = 30

Thus, the first five multiples of 6 are 6, 12, 18, 24 and 30.

**Exercise 1**. Write all the factors of the following numbers:

- (a) 21 (b) 23 (c) 24 (d) 27 (e) 36 (f) 72
- (g) 100 (h) 144 (i) 181 (j) 168 (k) 729

**Exercise 2**. Write first five multiples of (a) 5 (b) 8 (c) 9

**Exercise 3**. Find all the multiples of 9 up to 100.

**Exercise 4**. Which of the numbers (a) 15625 and (b) 151230 have 15 as their factor?

**Exercise 5**. Without actual division show that

- 17 is a factor of 17017.
- 19 is a factor of 190019.
- 23 is a factor of 2300023.

**Exercise 6**. Write the odd multiples of 19, less than 150.

**Exercise 7**. Write the even multiples of 23, less than 200.

**Exercise 8**. What number am I?

I am a two-digit multiple of 50.

**Exercise 9**. What number am I?

- I am an even number less than 100.
- One of my factors is 13.
- My digits are 1 apart.

Interesting & Mind-Stretching Set Of Problems!

One can enjoy while Solving!!

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