GCF's job: simplifying a single fraction
Simplifying 8/12 to lowest terms divides both numerator and denominator by their GCF, 4, giving 2/3 — the exact same value, written with smaller numbers. GCF answers "what's the smallest way to write this one fraction?"
LCM's job: adding two fractions with different denominators
Adding 1/4 and 1/6 needs a common denominator first — and the LCM of 4 and 6, which is 12, is the smallest one that works. Converting both fractions to twelfths: 1/4 becomes 3/12, and 1/6 becomes 2/12, so 1/4 + 1/6 = 3/12 + 2/12 = 5/12. LCM answers "what's the smallest shared denominator for combining two different fractions?"
Why both concepts trace back to the same underlying idea
GCF finds what two numbers share in common; LCM finds the smallest number both divide into. They're complementary views of the same relationship between two numbers — which is exactly why fraction simplification (needing GCF) and fraction addition (needing LCM) are really two different applications of one shared piece of number theory, not two unrelated formulas.