AMC8 2009

AMC8 2009 · Q17

AMC8 2009 · Q17. It mainly tests Primes & prime factorization, Perfect squares & cubes.

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 $y$?

正整数$x$和$y$是满足360与$x$的乘积为完全平方数且360与$y$的乘积为完全立方数的最小的两个正整数。求$x$和$y$的和。

(A) 80 80

(B) 85 85

(D) 165 165

(E) 610 610

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Factor 360 into $2\cdot2\cdot2\cdot3\cdot3\cdot5$. First increase the number of each factor as little as possible to form a square: $2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot5\cdot5=(2\cdot2\cdot2\cdot3\cdot3\cdot5)(2\cdot5)=(360)(10)$, so $x$ is 10. Then increase the number of each factor as little as possible to form a cube: $2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot5\cdot5\cdot5=(2\cdot2\cdot2\cdot3\cdot3\cdot5)(3\cdot5\cdot5)=(360)(75)$, so $y$ is 75. The sum of $x$ and $y$ is $10+75=85$.

答案（B）：将 $360$ 分解质因数为 $2\cdot2\cdot2\cdot3\cdot3\cdot5$。先尽量少地增加各质因数的个数，使其成为完全平方数：$2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot5\cdot5=(2\cdot2\cdot2\cdot3\cdot3\cdot5)(2\cdot5)=(360)(10)$，所以 $x=10$。再尽量少地增加各质因数的个数，使其成为完全立方数：$2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot5\cdot5\cdot5=(2\cdot2\cdot2\cdot3\cdot3\cdot5)(3\cdot5\cdot5)=(360)(75)$，所以 $y=75$。$x$ 与 $y$ 的和为 $10+75=85$。

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