Factors and Multiples
A factor divides into a number exactly; a multiple is what you get when you multiply it. The two ideas are mirror images, and the greatest common factor and lowest common multiple sit at opposite ends of the same relationship.
Formulas
LCM(a, b) = a × b ÷ GCF(a, b)
For a list: fold the pairwise rule left to right
GCF × LCM = a × b (two numbers only)
The identity GCF × LCM = a × b holds for exactly two numbers and is the quickest way to get one from the other. It does not extend to three or more, which is why longer lists are handled by applying the pairwise rule repeatedly rather than by multiplying everything together.
The Prime Factorisation Method
Breaking each number into primes makes both answers visible at once, which is why it is the method taught in school even though Euclid's algorithm is faster.
For the GCF, take the lowest power of each shared prime
12 = 2² × 3 and 18 = 2 × 3². They share a 2 and a 3, at the lowest powers 2¹ and 3¹, so the GCF is 6.
For the LCM, take the highest power of every prime that appears
From the same pair, the highest powers are 2² and 3², so the LCM is 4 × 9 = 36. Every prime in either number must appear, at the largest exponent seen.
A GCF of 1 means the numbers are coprime
They share no prime at all, so their LCM is simply their product. 8 and 15 are coprime despite neither being prime itself.
Where You Actually Need Them
The LCM is the lowest common denominator when adding fractions — it is why 1/12 + 1/18 is rewritten over 36ths rather than over 216ths. It also answers cycle questions: two events repeating every 12 and 18 days next coincide after 36 days.
The GCF is the tool for dividing things into equal groups without leftovers, and for reducing fractions and ratios to their simplest form. If you have 24 of one item and 36 of another and want identical bundles with nothing left over, the GCF of 12 tells you the largest number of bundles you can make.