   SEARCH HOME Math Central Quandaries & Queries  Question from Anonymous, a student: Two numbers have LCM = 60. If their product is 180, what is their HCF? Hi,

I'm going to illustrate with some larger numbers. Let m = 150 and n = 375. I want to find the LCM and HCF of m and n. First I find the prime factorizations of m and n.

m = 150 = 2 × 3 × 5 × 5
n = 375 = 3 × 5 × 5 × 5

It's easy to find a common multiple of m and n, m × n is a multiple of m and n.

m = 2 × 3 × 5 × 5               n = 3 × 5 × 5 × 5

m × n = 2 × 3 × 5 × 5 × 3 × 5 × 5 × 5

The least common multiple of m and n is

LCM = 2 × 3 × 5 × 5 × 5

Notice what is different between m × n and the LCM of m and n. The difference is one three and two fives. But 3 × 5 × 5 is the HCF of m and n. So what I see is that

m × n = LCM × HCF.

This is true for any positive integers and hence if you know m, n and their LCM you can find their HCF by division.

Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.