Page 134 - Ganit Kaushal

a = b × q
↓ ↓ ↓
Multiple Factor Factor
Also, in this situation, ‘a’ is said to be exactly (evenly) divisible by ‘b’ as well as by ‘q’.

Now, we can simply define that:

‘Factors’ of a number refer to all those numbers that evenly divide the number. These are also called
‘Divisors’ of that number, and that number is known as the ‘Multiple’ of each of its factors.

We have, 33 = 3 × 11 and this is read in many ways as follows:
↓ ↓ ↓
Multiple Factor Factor

‘3 is a factor of 33’ or ‘11 is a factor of 33’ or ‘33 is a multiple of 3’ or ‘33 is a multiple of 11’ or ‘33 is
exactly divisible by 3’ or ‘33 is exactly divisible by 11’.

The number is called a Prime Number if it has only two factors, 1 and itself, e.g., 13 = 1 × 13, so 13 is a
prime number, since it has 1 and 13 as its only factors. (We will study prime numbers in detail in section 5.3
of this chapter.)
A multiple of a number can be obtained by multiplying it by natural numbers 1, 2, 3, 4, 5, and so on.

For example: Multiples of 11 are 1 × 11, 2 × 11, 3 × 11, 4 × 11, … .
Multiples of 3 are 1 × 3, 2 × 3, 3 × 3, 4 × 3, 5 × 3, 6 × 3, … .

Real Life Applications and Properties of Factors and Multiples

Diwali is a festival of lights. We light ‘diyas’ at home and arrange them in rectangular form to look beautiful
on the festival of Diwali. Suppose we have to arrange 12 diyas in rectangular form with rows and columns.
How many ways can we do it?
The number of distinct ways 12 is factorised, gives the answer. Factors represent the number of rows
and columns of the rectangular arrangement. (see the pictures below)
(i) 1 × 12 or 12 × 1 arrangement.

(ii) 2 × 6 or 6 × 2 arrangement.
Note
Distinct Factors of 12 are 1 × 12,
(iii) 3 × 4 or 4 × 3 arrangement. 2 × 6, and 3 × 4. There are only 3

distinct ways we can factorise 12. So,
12 diyas can be arranged in 3 ways.

S. No. Properties of Factors Properties of Multiples
1. The factor of a number exactly divides the number, does not The multiple of a number is exactly divisible by the number.
leave any remainder. e.g., 4 is a factor of 12, as it exactly e.g., 12 is a multiple of 4, as 12 is exactly divisible by 4, and
divides 12, and leaves no remainder. leaves no remainder.
2. Every factor is smaller than or equal to the number. Every multiple is greater than or equal to the number.
3. The number of factors is finite. The number of multiples is infinite.
4. Every number is the highest factor of itself. Every number is the smallest multiple of itself.

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