AMC 10 · 2024 · #18
Grade 8 arithmeticPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Every integer either shares the prime 5 with 125 or it does not — Tool #16 (Change Focus) reframes the problem by that dichotomy and turns one hard question into two easy ones. Tool #7 (Identify Subproblems) then handles each case: (a) n coprime to 125 (no factor of 5) — here Euler's theorem nails n¹⁰⁰ ≡ 1 (mod 125) in one line because φ(125) = 100; (b) n is a multiple of 5 — then n¹⁰⁰ = 5¹⁰⁰ k¹⁰⁰, divisible by 5¹⁰⁰ ≫ 5³ = 125, so the remainder is 0. Tool #9 (Solve an Easier Related Problem) checks the coprime conclusion on a small case (n = 2) before trusting the general theorem, so the answer is grounded, not hand-waved.
125 = 5³, so only the prime 5 matters: every integer n is either coprime to 125 or a multiple of 5.
Choosing the dividing line by the prime factorization of 125 is the Grade 6 GCF/LCM mindset — only 5 matters.
6.NS.B.4Count The ComplementCase A — 5 ∤ n: since φ(125) = 100 and the exponent is 100, Euler's theorem gives n¹⁰⁰ ≡ 1 (mod 125).
When the exponent matches φ of the modulus, the power lands on 1 for every coprime base — a clean Grade 8 integer-exponent fact.
8.EE.A.1Identify SubproblemsCheck n = 2: 2¹⁰ ≡ 24 and 24⁵ ≡ -1, so 2¹⁰⁰ ≡ (-1)² = 1 (mod 125) — Case A confirmed.
A small-case check on n = 2 confirms the general Euler conclusion without trusting a theorem name in the dark.
8.EE.A.1Solve An Easier Related ProblemCase B — 5 ∣ n: write n = 5m, then n¹⁰⁰ = 5¹⁰⁰ m¹⁰⁰ is a multiple of 5³ = 125, so the remainder is 0.
A single factor of 5³ inside 5¹⁰⁰ already eats the modulus — the rest just goes along for the ride.
8.EE.A.1Identify SubproblemsEvery integer n lands in exactly one case, so the possible remainders are just {0, 1} — 2 distinct values → (B).
The complement-style split lined up so each side contributes exactly one remainder — Grade 5 "write the answer as a small set".
5.OA.A.2Count The ComplementThis AMC 10 problem only needs the Grade 8 integer-exponent rule and a clean "is n a multiple of 5 or not?" split — and the answer set turns out to be just {0, 1}!