Greatest common factor and least common multiple
Two numbers share some factors and some multiples. How do you find the biggest factor and the smallest multiple they share?
Where you are 3 inside
What this area is about
The greatest common factor of two positive whole numbers is the largest number that divides both. The least common multiple is the smallest positive number that both divide. The first answers questions about splitting things into identical groups; the second answers questions about when two repeating things line up again. For 4 and 6 they are 2 and 12.
How these fit together
By listing. Greatest common factor by listing writes out the factors of each number and takes the largest they share. Least common multiple by listing writes out multiples of each and takes the first they share.
By prime factors. Finding them from prime factorizations compares the powers of each prime in the two factorizations, which is faster when listing would run long, as with 72 and 84.
A path through it
Learn both listing methods first, in either order, then the prime factorization method.
These two ideas come back in common denominators, in factoring out the greatest common factor, and in the Euclidean algorithm.