2014 AMC 10A Problems/Problem 8

Problem

Which of the following number is a perfect square?

$\textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2$

Solution

Note that $\dfrac{n!(n+1)!}{2}=\dfrac{(n!)^2*(n+1)}{2}=(n!)^2*\dfrac{n+1}{2}$ . Therefore, the product will only be a perfect square if the second term is a perfect square. The only answer for which the previous is true is $\dfrac{17!18!}{2}=(17!)^2*9$ .

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2014 AMC 10A Problems/Problem 8

Problem

Solution