AMC 8 · 2016 · #15
Grade 6 number-theoryalgebraPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Computing 13⁴ - 11⁴ = 13920 directly is doable but messy, and we then still have to factor 13920. Tool #7 (Identify Subproblems) gives a cleaner path: use the difference-of-squares identity a² - b² = (a-b)(a+b) twice to break 13⁴ - 11⁴ into three small factors — (13-11), (13+11), and (13² + 11²) — and count the 2s in each factor separately. Tool #9 (Easier Related Problem) backs this up: testing a² - b² = (a-b)(a+b) on a tiny case like 5² - 3² makes the identity obvious, so the factoring move is trustworthy. We avoid Tool #13 (Algebra) and brute-force division because the factored form does the work for us.
Test the rule a² - b² = (a-b)(a+b) on a tiny case: 5² - 3² gives 16 both ways, so it is safe to use on 13 and 11.
Verifying an identity on small numbers first is the Tool #9 "easier problem" move — it builds confidence before applying the rule to harder numbers.
6.EE.A.1Solve An Easier Related ProblemSee 13⁴ and 11⁴ as (13²)² and (11²)², then apply the identity once to split it into (13² - 11²)(13² + 11²).
Splitting one hard expression into two easier ones is the core Tool #7 "subproblems" move.
6.EE.A.1Identify SubproblemsThe first factor 13² - 11² is itself a difference of squares, so split it again: (13-11)(13+11) = 2 × 24.
Re-applying the same easier-problem split to a leftover piece is exactly how subproblems compound — each call peels off a layer.
6.EE.A.1Identify SubproblemsThe third factor is a plain sum: 13² + 11² = 169 + 121 = 290, so the full product is 2 × 24 × 290.
Each factor is now a number we can hold in our head, so the hard expression has shrunk to three friendly pieces.
5.NBT.B.5Identify SubproblemsCount the 2s per factor: 2 = 2¹, 24 = 2³ × 3, 290 = 2¹ × 145. Add exponents 1 + 3 + 1 = 5, so the largest power of 2 is 2⁵.
Prime factorization tells you each factor's "2-content"; the exponent rule 2^a × 2^b = 2^a+b lets you total them in one step.
6.NS.B.4Identify SubproblemsThis AMC 8 problem only needs the Grade 6 idea of prime factorization plus one factoring trick — a² - b² = (a-b)(a+b) — that you can verify with tiny numbers like 5² - 3²!