Multiple-of Decode

Definition

Multiple-of Decode in plain English

Use Multiple-of Decode for divisibility, factors, primes, and GCD/LCM structure. Clues include multiple of, factor of, or prime factorization. Write the prime factorization; translate each condition into exponent floors and ceilings. Guessing multiples misses exponent constraints that factorization makes obvious on the page. Once the primes are written, GCD is min exponents and LCM is max exponents. Micro-example: n is a multiple of 12 and of 18. 12 = 2^2 · 3, 18 = 2 · 3^2, so n needs at least 2^2 · 3^2 = 36 as a factor. The LCM exponents are the decode, not a blind product of 12 and 18. If the stem mixes GCD and LCM language, decode each phrase into min or max exponents before multiplying anything back together.

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