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I came upon the following question in a recent district math test, and I have no clue how to solve it, besides using a calculator and doing some serious multiplication, but no calculators were available during the test.

Which of the following numbers is a perfect square?

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

Could somebody explain how to solve this problem step by step?

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  • $\begingroup$ D is such one jon $\endgroup$
    – Semsem
    Feb 5, 2014 at 20:44
  • $\begingroup$ @Semsem How did you get that answer? $\endgroup$ Feb 5, 2014 at 20:45
  • $\begingroup$ AMC10 by any chance? :3 $\endgroup$
    – qwr
    Feb 5, 2014 at 20:46

1 Answer 1

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(D) is equal to $\frac{17!17! \times 18}{2} = 17!17! \times 9 = (17!\times 3)^2$

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  • $\begingroup$ And the others can't be split the same way. $\endgroup$
    – vonbrand
    Feb 5, 2014 at 22:18

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