Page 135 - Ganit Kaushal

Ex 1. Write all the factors of the following numbers:
(a) 24 (b) 15 (c) 21
Sol. We know that the factor of a number is an exact divisor of the number.

(a) 24 = 1 × 24, 24 = 2 × 12, 24 = 3 × 8, 24 = 4 × 6
\ Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Pictorial representation
of factors of 24
(Follow the arrows alongside the pictorial representation to get all
the factors) 24 = 1 × 24
(b) 15 = 1 × 15, 15 = 3 × 5 = 2 × 12
\ Factors of 15 are 1, 3, 5 and 15. = 3 × 8
(c) 21 = 1 × 21, 21 = 3 × 7 = 4 × 6

\ Factors of 21 are 1, 3, 7 and 21.
Ex 2. Write the first five multiples of
(a) 5 (b) 8
Sol. A number is a multiple of each of its factors. So, multiply both numbers by the first five counting numbers.
(a) 5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20, 5 × 5 = 25.
\ So, 5, 10, 15, 20, 25 are the first five multiples of 5.
(b) 8 × 1 = 8, 8 × 2 = 16, 8 × 3 = 24, 8 × 4 = 32, 8 × 5 = 40.
So, the first five multiples of 8 are 8, 16, 24, 32 and 40.
Ex 3. Find all the multiples of 9 up to 100.
Sol. To find the multiples of 9 up to 100, we multiply Remember
it by the counting numbers 1, 2, 3, ..., 11. 1. Every number essentially has at least two
9 × 1 = 9, 9 × 2 = 18, 9 × 3 = 27, 9 × 4 = 36, factors: 1 and itself.
9 × 5 = 45, 9 × 6 = 54, 9 × 7 = 63, 9 × 8 = 72, 2. Every number is a multiple as well as a
9 × 9 = 81, 9 × 10 = 90, 9 × 11 = 99. factor of itself.

Hence, all multiples of 9 up to 100 are 9, 18, 27, 3. Division by zero is not defined. So, 0 is
36, 45, 54, 63, 72, 81, 90 and 99. never a factor of any number.
4. 0 is divisible by every natural number.
Common Factors and Highest Common Factor

Recall that the factors of a number are those numbers that divide it exactly. So, we write
Factors of 16 = 1, 2, 4, 8, 16 and Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.
Observe that the numbers 1, 2, and 4 are present in the factor lists of both 16 and 36. Therefore, 1, 2, and 4 are
called the common factors of 16 and 36.
Therefore, common factors of two or more numbers are the numbers that exactly divide all of them.

It can be noted that 1 is a common factor of all the natural numbers.
The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides each of
them exactly i.e., HCF of two or more numbers is the greatest of all the common factors. It is also called the
greatest common divisor (GCD).

For example: HCF (16, 36) = 4, since it is the greatest of all common factors 1, 2, 4 of 16 and 36.

Algorithm to Find HCF by Common Division
Step 1: Write all the given numbers in a row.

Step 2: Divide all the given numbers by a common divisor and get the quotients.

Prime Time 133