AMC 10 A 2019 Problem

Algebra Level 3

For how many integers n n between 1 and 50 inclusive is ( n 2 − 1 ) ! ( n ! ) n \frac{(n^2-1)!}{(n!)^{n}} an integer? (Recall that 0!=1.)

Source : AMC 10 A 2019 Question Paper


The answer is 34.

Gamerrk 1004 by gamerrk 1004 , 15, India

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1 solution

Direct computation: Length [ Table [ If [ ( n 2 − 1 ) ! ( n ! ) n ∈ Z , n , Nothing ] , { n , 50 } ] ] ⇒ 34 \text{Length}\left[\text{Table}\left[\text{If}\left[\frac{\left(n^2-1\right)!}{(n!)^n}\in \mathbb{Z},n,\text{Nothing}\right],\{n,50\}\right]\right] \Rightarrow 34 .

The integers are: 1,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,50.

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Is there a better solution?

Matteo Bianchi - 1 year, 10 months ago

For me, this was the easiest way. I'm retired. I was a computer generalist by trade.

A Former Brilliant Member - 1 year, 10 months ago

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