Least, but not Last!

Class 7 Maths · Chapter 11: Finding Common Ground · NCERT Ganita Prakash section P2 3.2

Learn

Kabir makes torans from 6 cm strips and Zoya from 8 cm strips. To make torans of the same length, the length must be a multiple of 6 (6, 12, 18, 24 …) and of 8 (8, 16, 24 …). The shortest is 24 cm. Common multiples go on forever (24, 48, 72 …), so there is no largest.

LCM(6, 8) = 24

This is the Lowest Common Multiple (LCM). A sweet shop gives free gajak every Monday (every 7 days), and Ishaan visits every 10 days. Starting on a Monday, they next coincide after LCM(7, 10) = 70 days. In the idli–vada game with 4 and 6, the first "idli-vada" is 12.

LCM(7, 10) = 70

LCM from primes: a multiple contains all the primes of the number. So a common multiple must contain both factorisations. 14 = 2 × 7 and 35 = 5 × 7: the smallest number containing both is 2 × 5 × 7 = 70. Take every prime that appears, each the most times it appears in either number.

LCM(14, 35) = 2 × 5 × 7 = 70

96 = 2 × 2 × 2 × 2 × 2 × 3 and 360 = 2 × 2 × 2 × 3 × 3 × 5. The LCM needs five 2s, two 3s and one 5: 2⁵ × 3² × 5 = 1440. Compare: the HCF takes the fewest of the common primes, the LCM the most of all primes.

LCM(96, 360) = 1440

Explore: common multiples.

Tap the numbers that are multiples of both 6 and 8.

Remember

LCM = the lowest common multiple; it answers "when do they meet again" problems.
From prime factorisations: every prime, each the most times it appears.

Quick check

What is the LCM of 36 and 54?

108
18
1944
216

Answer

108. 36 = 2² × 3², 54 = 2 × 3³: LCM = 2² × 3³ = 108.

Think about it

A herd of fewer than 200 cows passes equally through 3 gates, then 5 gates, then 7 gates. How many cows are there?

One answer

The number is a common multiple of 3, 5 and 7. The LCM is 105, and the next one, 210, is too many. So 105 cows.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.