AMC 8 · 2020 · #19
Grade 4 number-theorycountingPick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The condition "divisible by 15" splits cleanly into two smaller, very different sub-conditions: divisible by 5 (a last-digit rule) and divisible by 3 (a digit-sum rule). Tool #7 (Identify Subproblems) lets us pin down one digit at a time — the 5-rule forces the value of a, then the 3-rule narrows down b. Once both digits are constrained, Tool #2 (Make a Systematic List) finishes the job: write the few remaining candidate numbers in order and count them. We deliberately avoid Tool #13 (Algebra) — divisibility rules and listing handle this with no equations needed.
A five-digit flippy number must read ababa — positions 1, 3, 5 are a and positions 2, 4 are b — with a ≠ 0 and a ≠ b.
Reading a multi-digit number by its place-value positions is exactly the Grade 4 "read and write multi-digit whole numbers" skill.
4.NBT.A.2Identify SubproblemsDivisibility by 5 needs the last digit 0 or 5; that last digit is a and a ≠ 0, so a = 5 and the number is 5b5b5 (b ≠ 5).
Recognizing multiples of 5 by the last digit is part of the Grade 4 "factors and multiples" cluster.
4.OA.B.4Identify SubproblemsThe digits of 5b5b5 sum to 15 + 2b; since 15 is a multiple of 3, we only need 2b to be a multiple of 3 too.
The "sum-of-digits divisible by 3" rule is a standard Grade 4 divisibility test that lives in the factors-and-multiples standard.
4.OA.B.4Identify Subproblems2b is a multiple of 3 exactly when b is, so b ∈ {0, 3, 6, 9}, and every one of these already satisfies b ≠ 5.
Listing the one-digit multiples of 3 is direct Grade 4 multiples reasoning.
4.OA.B.4Identify SubproblemsList them in order — 50505, 53535, 56565, 59595 — giving 4 flippy multiples of 15, which is choice (B).
Generating every number from a clear rule ("5b5b5 for each allowed b") and counting matches the Grade 4 "generate a pattern following a rule" standard.
4.OA.C.5Make A Systematic ListThis AMC 8 problem only needs Grade 4 divisibility rules — "ends in 0 or 5" for 5, and "digits add up to a multiple of 3" for 3 — that you already know!