Hi Peter,

The least common denominator LCD of two fractions is the least common multiple LCM of the denominators. So I am going to compare the greatest common factor GCF and LCM of two positive integers. The words themselves tell you want they are

greatest common factor
least common multiple

Lets look at the GCF first (sometimes called the greatest common divisor GCD). Consider the numbers 36 and 60. I am looking for common factors of these two numbers, that is positive integers that divide both of them. 1 divides them both, so does 2 and 3 and 6 but what is the greatest positive integer that divides them both? I can answer this if I write the prime factorization of both 36 and 60.

36 = 2² × 3²
60 = 2² × 3 × 5

So 2² divides them both, 3 divides them both but not 3². 5 divided 60 but not 36. No other prime number divides them. Thus the greatest positive integer that divided them both is 2² × 3 = 12. Thus

GCF(36, 60) = 12

What about the least common multiple LCM of say 12 and 9? This time we want a multiple of 12 and 9. Certainly 12 × 9 = 108 is a multiple of them both but we want the least common multiple so is there a multiple which is less than 108? Again look at the prime factorizations.

12 = 2² × 3
9 = 3²

This time I can see that a multiple of 12 is

2² × 3 × (perhaps other factors)

and a multiple of 9 is

3² × (perhaps other factors)

Thus a multiple of both of them is

2² × 3² × (perhaps other factors)

and hence the least common multiple is

LCM(9, 12) = 2² × 3² = 36.

I hope this helps,
Penny