AMC 10 · 2024 · #4
Grade 6 number-theoryPick an answer.
The condition 2024 ∣ n! looks intimidating until we split it. Tool #7 (Identify Subproblems) says: factor 2024 into primes, then check each prime separately — n! is a multiple of 2024 exactly when n! contains every prime power in that factorization. The largest prime in the factorization sets a hard floor on n. Tool #3 (Eliminate Possibilities) then walks the answer choices: any candidate smaller than that floor cannot work, so they drop out immediately and the smallest candidate that clears the floor is the answer.
Factor 2024
2024 is eight times eleven times 23.
Recognizing 253 = 11 × 23 is the Grade 4 "find factor pairs" move; both 11 and 23 are prime.
4.OA.B.4Identify SubproblemsThe biggest prime rules
To see 23, n must reach at least 23.
Asking when a factorial contains a given prime is the same idea as asking for the LCM-style "smallest container" — a Grade 6 GCF/LCM mindset.
The largest prime in the factorization is what sets the floor on how far the count must climb.
▸ Why?
Every number has exactly one prime recipe, so that prime has to appear somewhere in the product.
▸ Why?
A prime cannot appear before the count reaches it, so anything smaller is ruled out at once.
Check the powers of two
By 8 the factor eight is already there.
Adding the exponents of like bases is the Grade 6 exponent rule 2^a · 2^b = 2^a+b — and 7 ＞ 3, so the 2-part is comfortable.
6.EE.A.1Identify SubproblemsRead the smallest n
The least n that satisfies all of them is 23.
Once the largest prime sets the floor, eliminating every smaller choice is a one-line check — the classic "largest prime is the bottleneck" move for least-multiple problems.
6.NS.B.4Eliminate PossibilitiesSplitting 2024 = 2³ × 11 × 23 turns this AMC 12 problem into a Grade 6 question — once you spot the biggest prime 23, the answer has nowhere left to hide.