Three numbers are in the ratio 3 : 2 : 13 and their LCM is 6318. Their HCF is:

Concept

When numbers are given as a ratio whose terms are pairwise coprime, each number is the same common multiplier times its ratio term, and that common multiplier is the HCF. So if the numbers are h·a, h·b and h·c with a, b and c pairwise coprime, then their HCF is h and their LCM is h × a × b × c, because no two ratio terms share a prime factor.

Application

1. The ratio terms 3, 2 and 13 are pairwise coprime, so the numbers can be written as 3h, 2h and 13h, where h is their HCF.

2. Take the common multiplier out of the LCM: LCM(3h, 2h, 13h) = h × LCM(3, 2, 13) = h × 3 × 2 × 13 = 78h.

3. The LCM given in the question is 6318, so 78h = 6318.

4. Divide to get the common multiplier: h = 6318 ÷ 78 = 81.

Cross-check

With h = 81 the three numbers are 243, 162 and 1053, whose prime factorisations are:

• 243 = 35

• 162 = 2 × 34

• 1053 = 34 × 13

The highest power of 3 present in all three numbers is 81, while 2 divides only one of them and 13 divides only one of them, so the greatest common divisor is 81. Multiplying the highest power of every prime that appears gives 6318, which is exactly the LCM stated in the question.

Hence the HCF of the three numbers is 81.