NCERT Solutions for Class 7th Maths Chapter 3 Figure it Out — Finding the HCF of Numbers Using Prime Factorisation

Book page 54 Updated on2026-09-19

Q1.

Find the HCF of the following numbers: (a) 24, 180 (b) 42, 75, 24 (c) 240, 378 (d) 400, 2500 (e) 300, 800

Answer

Factorise, mark the shared primes, take the minimum count of each.

	Prime factorisations	Shared primes (minimum count)	HCF
(a)	24 = 2 × 2 × 2 × 3 180 = 2 × 2 × 3 × 3 × 5	two 2s, one 3	2 × 2 × 3 = 12
(b)	42 = 2 × 3 × 7 75 = 3 × 5 × 5 24 = 2 × 2 × 2 × 3	one 3 only (75 has no 2)	3
(c)	240 = 2 × 2 × 2 × 2 × 3 × 5 378 = 2 × 3 × 3 × 3 × 7	one 2, one 3	2 × 3 = 6
(d)	400 = 2 × 2 × 2 × 2 × 5 × 5 2500 = 2 × 2 × 5 × 5 × 5 × 5	two 2s, two 5s	2 × 2 × 5 × 5 = 100
(e)	300 = 2 × 2 × 3 × 5 × 5 800 = 2 × 2 × 2 × 2 × 2 × 5 × 5	two 2s, two 5s	2 × 2 × 5 × 5 = 100

Checks:
(a) 24 ÷ 12 = 2, 180 ÷ 12 = 15 ✓
(b) 42 ÷ 3 = 14, 75 ÷ 3 = 25, 24 ÷ 3 = 8 ✓
(c) 240 ÷ 6 = 40, 378 ÷ 6 = 63 ✓
(d) 400 ÷ 100 = 4, 2500 ÷ 100 = 25 ✓
(e) 300 ÷ 100 = 3, 800 ÷ 100 = 8 ✓

Tip: in (b), 75 is odd, so 2 cannot be in the HCF however many 2s the other two numbers have. Scanning for a number that lacks a prime is the fastest way to throw that prime out.

Q2.

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Answer

No, one cannot say that. The splittings 6 × 12 and 8 × 18 share no visible factor, but the numbers themselves share plenty.

72 = 6 × 12 = (2 × 3) × (2 × 2 × 3) = 2 × 2 × 2 × 3 × 3
144 = 8 × 18 = (2 × 2 × 2) × (2 × 3 × 3) = 2 × 2 × 2 × 2 × 3 × 3
Shared: three 2s and two 3s
HCF = 2 × 2 × 2 × 3 × 3 = 72

In fact 144 = 72 × 2, so 72 divides 144 and is itself the HCF. The common factors are all twelve factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

Why it happens: 6, 12, 8 and 18 are composite, so each still hides primes inside it. 6 and 8 look unrelated, but both contain a 2. Only when every piece has been broken down to primes can you trust what you see. Comparing half-finished factorisations is exactly the mistake this chapter's method is designed to prevent.

Check it yourself: 6 and 8 already share the factor 2, and 12 and 18 share 6. Even the given splittings are not really factor-free.

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