AMC 10 · 2024 · #6
Grade 8 geometry-2dPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
For a fixed product, two positive numbers whose sum is smallest are the pair closest to the square root. Since √(2024) ≈ 45, we only need to test integers near 45 — a perfect setup for Tool #6 (Guess and Check) starting from the square-root estimate and adjusting up or down. Tool #7 (Identify Subproblems) splits the work into two pieces — first find the prime factorization of 2024 so we know which integers near 45 are actually divisors, then turn the winning (ℓ, w) pair into a perimeter.
Subproblem A: prime-factor 2024 = 2³ × 11 × 23, so every side length is built from these primes.
Prime factorization lists every "building block" of 2024, so every divisor is built by choosing some of these primes — Grade 6 "factors and multiples".
6.NS.B.4Identify SubproblemsEstimate √2024 ≈ 45 (since 45² = 2025) to know where the balanced factor pair should sit.
The closer the two sides are, the smaller ℓ + w. So the best pair will sit on either side of 45 — Grade 8 square-root estimation.
8.EE.A.2Guess And CheckGuess near 45: split the primes into 46 × 44 = 2024, with both sides just 1 away from 45.
Trying numbers right next to √(2024) first is the Tool #6 "educated guess" move — Grade 4 factor-pair thinking.
4.OA.B.4Guess And CheckCheck the neighbor: the factor just below 44 is 23, pairing with 88 for sum 111 — above 90, so (44, 46) wins.
Once we step away from √(2024), the pair spreads apart fast — a quick "check the neighbor" rules it out.
4.OA.B.4Guess And CheckSubproblem B: plug (46, 44) into P = 2(ℓ + w) = 2 × 90 = 180 → choice (B).
The rectangle perimeter formula is Grade 4 measurement — just plug the winning sides in.
4.MD.A.3Identify SubproblemsThis AMC 10 problem only needs Grade 8 square-root estimation you already know — find the pair of factors closest to √(area) and the perimeter is forced!