Which of the following expressions is an integer?

Correct answer marked below

\sqrt{\frac{14!\cdot 15!}{2}}

\sqrt{\frac{15!\cdot 16!}{2}}

\sqrt{\frac{16!\cdot 17!}{2}}

\sqrt{\frac{17!\cdot 18!}{2}}

Correct

\sqrt{\frac{18!\cdot 19!}{2}}

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Comments

BrentAug 12

Took me a sec to see the pattern but once you rewrite the product it becomes clear. Nice question.

LamanAug 25

Same here. What helped me was factoring out the smaller factorial.

AyeshaAug 15

Not really faster, but once the pattern clicks you can kind of scan them. The middle one seemed most likely to me.

RupinderJun 26

Wait, why is 2 in the denominator a problem? Can't we just pull it out somehow?

MeenalJul 3

Think about the exponent of 2 in the factorial. It has to pair up with another factor of 2 to come out of the root.

SubhamJul 2

That's what tripped me up too. I kept forgetting the denominator also needs a matching factor.

SmritiJun 11

Got this one right but it took me almost 3 minutes. Definitely not a 30-second problem for me.

OliverJun 15

Same, I overthought it at first and started multiplying stuff out. Bad idea.

ZiggyAug 26

Is the trick here that consecutive factorials share a common factorial, so you get something like a perfect square times a leftover?

PrajwalAug 31

Yep, that's the way to do it. Then you just need the leftover to be a perfect square.

What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.