2014 AMC 10A Problem 8

Question

Which of the following numbers is a perfect square?

\dfrac{14!15!}2

\dfrac{15!16!}2

\dfrac{16!17!}2

\dfrac{17!18!}2

\dfrac{18!19!}2

Write each choice as

\frac{n!(n+1)!}{2}

Only

\frac{n+1}{2}

needs to be a square

Note that all of these answer choices are of the form

\dfrac{n!(n + 1)!}{2} = \dfrac{(n!)^2(n + 1)}{2}.

We have that

(n!)^2

is square, so we need

\dfrac{n + 1}{2}

to be square as well.

This means that

n + 1

must be twice a perfect square. The only choice we have is

n + 1 = 18,

which gives us

n = 17.

Thus, D is the correct answer.

Problem 8 in other years