NCERT Solutions for Class 7th Maths Chapter 3 In-text Questions — Patterns, Properties, and a Pretty Procedure!

Book page 58–59 Updated on2026-09-19

Q1.

The HCF of 6 and 18 is 6, which is one of the two given numbers. Find more such number pairs where the HCF is one of the two numbers. How can we describe such pairs of numbers?

Answer

This happens exactly when one number is a factor of the other.

Pair	Relation	HCF
6, 18	18 = 6 × 3	6
5, 40	40 = 5 × 8	5
12, 60	60 = 12 × 5	12
9, 9	9 = 9 × 1	9
14, 98	98 = 14 × 7	14

The general statement: if one number is a factor of the other, their HCF is the smaller number — and the other number is a multiple of it.

Why it happens: the HCF of two numbers can never be bigger than the smaller number, since it has to divide it. So the best possible case is that the smaller number divides the larger one too. When it does, it is a common factor and it is as large as a common factor can be — so it is the HCF.

Did you know? A statement that holds in every possible case, like the one above, is called a general statement, and the process of arriving at it is generalisation.

Q2.

For number pairs satisfying this property (i.e., one of the numbers is the HCF), (a) if m is a number, what could be the other number? (b) if 7k is a number, what could be the other number?

Answer

The other number must be a multiple of the first.

(a) If one number is m, the other could be 2m, 3m, 4m, … — in general am, where a is any positive integer.

HCF of m and am = m
Example: m = 15 and a = 4 → 15 and 60, HCF = 15 ✓

(b) If one number is 7k, the other could be any multiple of 7k, that is 7ka for a positive integer a — for instance 14k, 21k, 28k.

HCF of 7k and 14k = 7k
Example: k = 3 → 21 and 42, HCF = 21 = 7 × 3 ✓

Why it happens: in the general form n and an, the prime factorisation of an contains every prime of n plus whatever a adds. So the largest subpart shared by the two is the whole of n, and HCF(n, an) = n. Writing it with letters shows that the pattern does not depend on which particular numbers you picked.

Tip: the other number could also be 7k itself (taking a = 1), since HCF(7k, 7k) = 7k. A number is always a factor of itself.

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