Greatest common factor and least common multiple

Two different questions

• The greatest common factor (GCF) of two numbers is the largest number that divides both of them. Some books call it the greatest common divisor (GCD).
• The least common multiple (LCM) is the smallest positive number that both of them divide into.

For small numbers, listing works fine. Take \(16\) and \(24\):

• Factors of \(16\): \(1, 2, 4, 8, 16\). Factors of \(24\): \(1, 2, 3, 4, 6, 8, 12, 24\). The biggest one they share is \(8\), so \(\text{GCF}(16, 24) = 8\).
• Multiples of \(16\): \(16, 32, 48, 64, \dots\) Multiples of \(24\): \(24, 48, 72, \dots\) The first one they share is \(48\), so \(\text{LCM}(16, 24) = 48\).

With bigger numbers the lists get long. Prime factorization is faster.

Using prime factorization

Relatively prime numbers

Two numbers are relatively prime when their GCF is \(1\). Neither number has to be prime: \(9 = 3^2\) and \(20 = 2^2 \cdot 5\) are both composite, but they share no prime factor.

When two numbers are relatively prime, their LCM is just their product: \(\text{LCM}(9, 20) = 180\).

Another quick case: if one number is a factor of the other, the smaller one is the GCF and the larger one is the LCM. For \(6\) and \(30\), the GCF is \(6\) and the LCM is \(30\).

GCF or LCM? Reading word problems

• Splitting things into the largest possible equal groups with nothing left over? That's a GCF problem.
• Looking for when two repeating events happen together again, or the smallest amount that can be split evenly in two ways? That's an LCM problem.

Practice