AMC 8 · 2016 · #9
Grade 5 number-theoryPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) splits the question into two clean jobs: (a) factor 2016 into primes, then (b) add the distinct prime bases. Tool #6 (Guess and Check) handles step (a) — trial-divide by the small primes 2, 3, 5, 7, … in order until the quotient becomes 1. We deliberately avoid heavier tools like #13 (Algebra) because plain trial division is the most direct path for a four-digit number.
Subproblem 1: factor 2016 by dividing out 2 repeatedly until the odd quotient 63 appears.
Repeated division by 2 is a Grade 5 multi-digit division skill — no special technique needed.
5.NBT.B.6Guess And Check63 is odd, so switch to 3 (digit sum 6+3=9 is a multiple of 3); dividing by 3 twice leaves 7.
Using the digit-sum rule for 3 and recognizing prime factors is exactly the Grade 4 "factors and multiples" standard.
4.OA.B.4Guess And Check7 is itself prime, so the factorization is complete: 2⁵ × 3² × 7.
Recognizing that 7 is prime ends the factorization — a Grade 4 prime/composite check.
4.OA.B.4Identify SubproblemsSubproblem 2: ignore the exponents and keep only the distinct prime bases 2, 3, and 7.
"Distinct" means we list each prime base once — a careful reading of the problem, not a calculation.
4.OA.B.4Identify SubproblemsAdd the three distinct primes 2 + 3 + 7 and match the result to a choice.
Adding three small whole numbers is a Grade 2 fluency skill.
2.NBT.B.5Identify SubproblemsThis AMC 8 problem only needs Grade 5 division and the Grade 4 idea of prime factors that you already know!
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