Problem 17 · 2009 AMC 8
Hard
Number Theory
exponent-parity-mod
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y?
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Answer: B — 85.
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Hint 1 of 2
A perfect square is a number whose prime exponents are ALL even; a perfect cube has all exponents divisible by 3. So prime-factor 360 first and just look at the exponents.
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Hint 2 of 2
To minimize the multiplier, top up each exponent to the NEXT even number (for a square) or next multiple of 3 (for a cube) — add only the missing primes, nothing extra.
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Approach: round each prime's exponent up to the target
- 360 = 2³ · 3² · 5¹. Read the exponents: 3, 2, 1.
- Square (all exponents even): 3 → 4 needs one more 2; 2 is already even; 1 → 2 needs one more 5. So multiply by 2 × 5 = 10. Thus x = 10.
- Cube (all exponents multiples of 3): 3 is fine; 2 → 3 needs one more 3; 1 → 3 needs two more 5's. So multiply by 3 × 5² = 75. Thus y = 75.
- x + y = 10 + 75 = 85.
- Why this transfers: "multiply to make a perfect square/cube" is purely about exponents — round each up to the next even / next multiple of 3, and you're done. The minimum multiplier supplies exactly the missing primes.
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