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Prime Factorisation to Check if One Number is Divisible by Another

We know that 72 is divisible by 18 because when we divide 72 by 18, the remainder is zero. Let us
explore the way that can be used to check if one number is divisible by another without carrying
out long division.
Consider the prime factorization of 72 and 18.
72 = 2 × 2 × 2 × 3 × 3 and 18 = 2 × 3 × 3.

All prime factors of 18 are also the prime factors of 72. Also, the prime factorisation of 18 is included
in the prime factorisation of 72.

Thus, we can say that if one number is divisible by another, the prime factorisation of the second
number is included in the prime factorisation of the first number.
Example 12: Is 160 divisible by 24?
Solution: Find the prime factorisations of both the numbers.

160 = 2 × 2 × 2 × 2 × 2 × 5 and 24 = 2 × 2 × 2 × 3.
As we can see the prime factor 3 is present in prime factorisation of 24 but not in prime factorisation
of 160. So, 160 is not divisible by 24.
Example 13: Is 98 divisible by 28?
Solution: Find the prime factorisations of both the numbers.

98 = 2 × 7 × 7 and 28 = 2 × 2 × 7.
All prime factors of 28 are also prime factors of 98. But the prime factorisation of 28 is not included
in the prime factorisation of 98. This is because 2 occurs twice in the prime factorisation of 28 but
only once in the prime factorisation of 98. This means that 98 is not divisible by 28.

Perfect, Abundant and Deficient Numbers

We have learnt about divisors/factors of a number. We also discussed about proper divisors.

• If the sum of proper divisors of a number is equal to the number itself, then it is called perfect
number.
For example: 6 and 28 are perfect numbers.
As, proper divisors of 6 = 1, 2 and 3

And, sum of proper divisors = 1 + 2 + 3 = 6
Proper divisors of 28 = 1, 2, 4, 7 and 14

Sum of its proper divisors = 1 + 2 + 4 + 7 + 14 = 28
Hence, the numbers 6 and 28 are perfect numbers.
• If the sum of proper divisors is greater than the number, it is called abundant number, while
if the sum of proper divisors is less than the number, it is called deficient number.

For example:
(i) 12 is an abundant number. Knowledge Desk
As, proper divisors of 12 = 1, 2, 3, 4 and 6 Number 12 is the first abundant

And, sum of its proper divisors = 1 + 2 + 3 + 4 + 6 = 16 > 12 number.
Hence, 12 is an abundant number.

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