NCERT Solutions for Class 7th Maths Chapter 3 Figure it Out — Patterns, Properties, and a Pretty Procedure!

Book page 59 Updated on2026-09-19

Q1.

Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold. (a) Two consecutive even numbers (b) Two consecutive odd numbers (c) Two even numbers (d) Two consecutive numbers (e) Two co-prime numbers. Share your observations with the class.

Answer

Try examples first, then state the rule.

	Examples	HCF	General statement
(a) consecutive even	6, 8 → 2 14, 16 → 2 30, 32 → 2	always 2	The HCF of two consecutive even numbers is 2.
(b) consecutive odd	7, 9 → 1 15, 17 → 1 21, 23 → 1	always 1	Two consecutive odd numbers are co-prime; their HCF is 1.
(c) two even	6, 8 → 2 12, 18 → 6 20, 100 → 20	an even number	The HCF of two even numbers is always even (at least 2), but it can be anything even.
(d) consecutive	8, 9 → 1 20, 21 → 1 99, 100 → 1	always 1	Two consecutive numbers are always co-prime.
(e) co-prime	4, 9 → 1 25, 42 → 1	always 1	The HCF of two co-prime numbers is 1, by definition.

Why they hold:

(d) Write them as n and n + 1. A common factor must divide the difference, which is 1. The only number that divides 1 is 1 itself.
(a) Write them as 2n and 2n + 2 = 2(n + 1). Both contain a 2, so 2 is common. Beyond that, the HCF of n and n + 1 is 1 by (d), so nothing more can be shared. HCF = 2 × 1 = 2.
(b) Their difference is 2, so any common factor divides 2 — it is 1 or 2. But both numbers are odd, so 2 cannot divide them. Only 1 is left.
(c) Both are multiples of 2, so 2 is a common factor and the HCF is at least 2. How much more they share depends on the numbers, so no fixed value can be promised.
(e) "Co-prime" means "no common prime factor". With no shared prime, the largest common subpart is empty, which is 1.

Q2.

The LCM of 3 and 24 is 24 (it is one of the two given numbers). (a) Find more such number pairs where the LCM is one of the two numbers. (b) Make a general statement about such numbers. Describe such number pairs using algebra.

Answer

(a) More such pairs:

Pair	Relation	LCM
3, 24	24 = 3 × 8	24
5, 20	20 = 5 × 4	20
7, 21	21 = 7 × 3	21
6, 36	36 = 6 × 6	36
11, 11	11 = 11 × 1	11

(b) General statement: the LCM is one of the two numbers exactly when one number is a factor of the other — and then the LCM is the larger number.

In algebra: the pair is n and an, where a is a positive integer
LCM(n, an) = an
HCF(n, an) = n

Why it happens: an is already a multiple of n and a multiple of itself, so it is a common multiple. No common multiple can be smaller than the larger number, because a multiple of an is at least an. So an is the lowest — the LCM.

Tip: this is the same family of pairs as in the HCF question on page 58. For such a pair, the smaller number is the HCF and the larger is the LCM — and indeed n × an = HCF × LCM ✓

Q3.

Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold. (a) Two multiples of 3 (b) Two consecutive even numbers (c) Two consecutive numbers (d) Two co-prime numbers

Answer

	Examples	General statement
(a) two multiples of 3	6, 9 → 18 12, 18 → 36 15, 21 → 105	The LCM is always a multiple of 3.
(b) consecutive even	6, 8 → 24 10, 12 → 60 14, 16 → 112	The LCM is half the product of the two numbers.
(c) consecutive	8, 9 → 72 11, 12 → 132 20, 21 → 420	The LCM is the product of the two numbers.
(d) co-prime	4, 9 → 36 7, 11 → 77 25, 42 → 1050	The LCM is the product of the two numbers.

Why they hold:

(a) The LCM is a multiple of each number, and each number is a multiple of 3. So the LCM contains a 3 in its prime factorisation.
(c) and (d) Co-prime numbers share no prime. So the LCM must take every prime of the first and every prime of the second, with nothing overlapping — which is exactly their product. Consecutive numbers are always co-prime, so (c) is a special case of (d).
(b) Two consecutive even numbers are 2n and 2(n + 1), with HCF 2. Their product is 4n(n + 1), while the LCM only needs one of the two 2s, giving 2n(n + 1) — half the product. Check with 6 and 8: product 48, LCM 24 ✓

Tip: all four statements are the same fact seen from different sides — HCF × LCM = product. When the HCF is 1, LCM = product; when the HCF is 2, LCM = half the product.

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