GCF & LCM

The GCF (greatest common factor) of two numbers is the largest number that divides into both. For 12 and 18 the shared factors are 1, 2, 3 and 6, so the GCF is 6. This is what you use to reduce fractions.

The LCM (least common multiple) is the smallest number both of them divide into. For 12 and 18 that is 36. This is what you use to find a common denominator.

Quick check on which one you need: GCF makes things smaller, LCM makes things bigger.

Both come directly out of prime factorizations. Write each number in canonical form and compare exponents prime by prime:

12 = 2²·3¹ 18 = 2¹·3²

Take the minimum exponent on each prime for the GCF (2¹·3¹ = 6) and the maximum for the LCM (2²·3² = 36). Any prime missing from a number counts as exponent 0.

This gives the identity GCF(a,b) · LCM(a,b) = a·b, since min(x,y) + max(x,y) = x + y for every exponent pair. Checking: 6 × 36 = 216 = 12 × 18. That identity is the fast route to an LCM once you have the GCF, and it is why the Euclidean algorithm (which finds GCF quickly without factoring) is useful for both.