AMC 10 · 2024 · #14
Grade 8 arithmeticPick an answer.
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.
Split into two families
Everything hinges on whether five divides n.
Choosing the dividing line by the prime factorization of 125 is the Grade 6 GCF/LCM mindset — only 5 matters.
Splitting the cases by the prime hiding in the modulus is what makes the two families behave differently.
▸ Why?
Every number has exactly one prime recipe, so whether it shares that prime is a clean yes or no.
▸ Why?
Numbers sharing the prime and numbers avoiding it obey different divisibility rules, so they split cleanly.
The coprime case
The totient is exactly 100.
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 SubproblemsSanity-check with an example
Two to the hundredth is also 1.
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 ProblemThe multiple-of-five case
Powers of five overflow, giving 0.
A single factor of 5³ inside 5¹⁰⁰ already eats the modulus — the rest just goes along for the ride.
8.EE.A.1Identify SubproblemsCount the possible values
Only zero and one, so 2 values.
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.2Change Focus Count The ComplementThis AMC 12 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}!