Square Factorials

Number Theory Level 1

Which of the following is a perfect square ?

Notation : $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

97!\times98!

98!\times99!

99!\times100!

100!\times101!

3 solutions

展豪張
Jun 1, 2016

Relevant wiki: Perfect Squares, Cubes, and Powers

$n!\times(n+1)!=(n+1)(n!)^2$ is a perfect square iff $n+1$ is a perfect square.
The only option is $99!\times 100!$ where $100$ itself is a perfect square.

That is absolutely correct.

Hana Wehbi - 5 years ago

This problem is already posted by me, @Hana Nakkache , here .

Anish Harsha - 5 years ago

I am sorry, I saw similar problem in one of my kids homework so I thought it would be nice to post it. If you want me to delete it, let me know.

Hana Wehbi - 5 years ago

No No . No need to delete. Let it be, it's a good problem.

Anish Harsha - 5 years ago

@Anish Harsha – Thank you so much.

Hana Wehbi - 5 years ago

Ashish Menon
Jun 5, 2016

$99! × 100! = 99! × 99! × 100 = {\left(99!\right)}^{2} × {10}^2 = {\left(10 × 99!\right)}^2$ .
So, $\color{#3D99F6}{\boxed{99! × 100!}}$ is a perfect square.

Absolutely correct.

Hana Wehbi - 5 years ago

Aniruddha Bagchi
Jun 11, 2016

It's 99! x 100!. Because we can rewrite it as 99! x 99! x 100 and that's a perfect square . Nice question , Hana. One must know to play with factorials to solve this question.

Thank you.

Hana Wehbi - 5 years ago

Square Factorials

3 solutions

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