Given that n is a positive integer, prove that (2n)! is divisible by 22n 1
Firstly we write, 22n 1 = 2n + n 1 = 2n 2n 1
Then, (2n)! = 1 2 3 ... (2n 1) 2n
Clearly, (2n)! is divisible by 2n, but as 2n 1 2n, one of the earlier factors of (2n)! must be 2n 1.
Hence (2n)! is divisible by 22n 1.