AMC10 2014 A

AMC10 2014 A · Q8

AMC10 2014 A · Q8. It mainly tests Perfect squares & cubes.

Which of the following numbers is a perfect square?

下列哪一个数是完全平方数？

(A) $\frac{14!15!}{2}$ $\frac{14!15!}{2}$

(B) $\frac{15!16!}{2}$ $\frac{15!16!}{2}$

(D) $\frac{17!18!}{2}$ $\frac{17!18!}{2}$

(E) $\frac{18!19!}{2}$ $\frac{18!19!}{2}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

Note that $\frac{n!(n+1)!}{2} = (n!)^2 \cdot \frac{n+1}{2}$, which is a perfect square if and only if $\frac{n+1}{2}$ is a perfect square. Only choice D satisfies this condition.

注意到 $\frac{n!(n+1)!}{2} = (n!)^2 \cdot \frac{n+1}{2}$，这个数是完全平方数当且仅当 $\frac{n+1}{2}$ 是完全平方数。只有选项 D 满足这个条件。

Topics

NT.006 Perfect squares & cubes