AMC 8 · 2009 · #17
Grade 8 number-theoryPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem looks like one question but is really two independent subproblems glued together by a final sum, so Tool #7 (Identify Subproblems) is the natural opener: find x, find y, then add. Both subproblems share the same setup: factor 360 into primes and look at the exponents. Tool #12 (Find a Pattern) names the pattern that runs through both — to make a number a perfect square (or cube), each prime exponent has to be bumped up to the next even number (or next multiple of 3). That single rule mechanically gives both x and y from the factorization of 360.
Factor 360 into primes to get 2³ · 3² · 5¹ — the exponents are 3, 2, 1.
Reading off prime exponents is the Grade 6 number-theory move that turns a square/cube question into a question about each exponent separately.
6.NS.B.4Identify SubproblemsFor a perfect square, raise each exponent to the next even number, so the missing factor is x = 10.
"Push every exponent up to the next even number" is the pattern (Tool #12) that defines a perfect square via prime factorization — exactly the Grade 8 reasoning that connects exponents to squares.
8.EE.A.2Draw A Venn DiagramCheck: 360 · 10 = 3600 = 60², a perfect square.
Confirming the square root explicitly is the quick sanity check that the exponent bookkeeping was right.
8.EE.A.2Identify SubproblemsFor a perfect cube, raise each exponent to the next multiple of 3, so the missing factor is y = 75.
Same pattern as before, swapping "next even" for "next multiple of 3" — the rule is one line longer than the work.
8.EE.A.2Draw A Venn DiagramCheck: 360 · 75 = 2³ · 3³ · 5³ = 30³ = 27000, a perfect cube.
Seeing the matching exponents collapse into (2 · 3 · 5)³ is the satisfying visual that the cube condition is met.
8.EE.A.2Identify SubproblemsAdd the two subproblem answers: x + y = 10 + 75 = 85 → (B).
The final "and now add" step is the close of the Tool #7 split — once both subproblems are solved, the rest is one-digit addition.
4.NBT.B.4Identify SubproblemsBreak 360 into primes, then bump each exponent up to the next even number for squares and the next multiple of 3 for cubes — that single pattern hands you both x and y.
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Same archetype — closest grade level first.
