# Playing With Numbers

#### 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 that

68 ÷ 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 that

36 ÷ 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.