2009 AMC 8 Problem 17

Question

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?

80

85

115

165

610

Factor

360

into primes.

Make exponents even for a square and multiples of

3

for a cube.

For a number to be a perfect square, every exponent in the prime factorization must be even. For it to be a cube, the exponents must be divisible by

3.

We can factor

360

to get

360 = 2^3 \cdot 3^2 \cdot 5.

For

360x

to be a perfect square and

x

to be minimized,

x

must have one factor of

2

and one factor of

5.

Therefore, we can let

x = 10.

For

360y

to be a cube,

y

must have one factor of

3

and two factors of

5.

Therefore, we can let

y = 75,

suggesting

x + y = 85.

Thus, B is the correct answer.

Problem 17 in other years