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Factorial Divisibility

Problem

Given that n is a positive integer, prove that (2n)! is divisible by 22n minus 1


Solution

Firstly we write, 22n minus 1 = 2n + n minus 1 = 2n 2n minus 1

Then, (2n)! = 1 times 2 times 3 times ... times (2n minus 1) times 2n

Clearly, (2n)! is divisible by 2n, but as 2n minus 1 less than 2n, one of the earlier factors of (2n)! must be 2n minus 1.

Hence (2n)! is divisible by 22n minus 1.

Problem ID: 40 (Mar 2001)     Difficulty: 3 Star

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