least common multiple
Two buses leave a stop together: one returns every 6 minutes, the other every 8 minutes. When will they next leave together? You need the first time that is a multiple of both 6 and 8 — that is 24 minutes. The smallest positive number that is a multiple of every given number is their least common multiple.
A direct method is to list multiples of each number until one appears in both lists. Multiples of 6 are 6, 12, 18, 24, …; multiples of 8 are 8, 16, 24, …; the first match is 24. For larger numbers, use prime factorization and take each prime to its highest power appearing in any number, then multiply.
The least common multiple is exactly what you need to find a common denominator when adding fractions with different denominators. There is also a neat shortcut: for two numbers, the product of the least common multiple and the greatest common factor equals the product of the numbers, so LCM(a, b) = (a × b) / GCF(a, b).
Find the LCM of 4 and 6. Primes: 4 = 2^2 and 6 = 2 × 3. Take highest powers: 2^2 × 3 = 12. So LCM(4, 6) = 12.
Take each prime to its highest power across the numbers, then multiply for the LCM.
The least common multiple of two numbers is always at least as large as the larger number, while the greatest common factor is always at most as large as the smaller one. They sit at opposite ends.