AMC 8 2009 Problem 17

AMC8 2009 年第 17 题

整除与因数★★★☆☆

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

x 和 y 是满足 360 和 x 之积为平方数，360 和 y 之积为立方数的最小正整数。那么 x 和 y 的和是多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Take the prime factorization of 360.

360=2^3*3^2*5

. We want

x

to be as small as possible. And you want x*360 to be a square. So x=2*5=10.

y

is similar.

y=3*5^2=3*25=75

So, x+y=75+10=85, or 85.

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