AMC10 2004 B

AMC10 2004 B · Q6

AMC10 2004 B · Q6. It mainly tests Basic counting (rules of product/sum), Perfect squares & cubes.

Which of the following numbers is a perfect square?

以下哪个数是完全平方数？

(A) 98! \cdot 99! 98! \cdot 99!

(B) 98! \cdot 100! 98! \cdot 100!

(D) 99! \cdot 101! 99! \cdot 101!

(E) 100! \cdot 101! 100! \cdot 101!

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Note that for $m<n$ we have $$m!\cdot n!=(m!)^2\cdot (m+1)\cdot (m+2)\cdots n.$$ Therefore $m!\cdot n!$ is a perfect square if and only if $$(m+1)\cdot (m+2)\cdots n$$ is a perfect square. For the five answer choices, that quantity is $$99,\ 99\cdot 100,\ 100,\ 100\cdot 101,\ \text{and }101,$$ and of those only $100$ is a perfect square. Therefore the answer is $99!\cdot 100!$.

（C）注意当 $m<n$ 时，有 $$m!\cdot n!=(m!)^2\cdot (m+1)\cdot (m+2)\cdots n.$$ 因此，$m!\cdot n!$ 是完全平方数当且仅当 $$(m+1)\cdot (m+2)\cdots n$$ 是完全平方数。对于五个选项，该量分别为 $$99,\ 99\cdot 100,\ 100,\ 100\cdot 101,\ \text{以及 }101,$$ 其中只有 $100$ 是完全平方数。因此答案是 $99!\cdot 100!$。

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