Composite or Prime
Number Theory
Level
1
If n is a composite number, then 2^{n} - 1 is which of them?
Prime
Composite
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1 solution
n is a composite , then let n = kl, where k,l are positive integers , k,l>1. Now. 2^{n} - 1 = 2^{kl} - 1= 2^{k}{l} - 1^{l}. We know, that for any positive integers x , y, (x^{y} - 1) is divisible by (x - 1 ). hence, 2^{n} - 1 is divisible by 2^{k} - 1 , which is greater than 1.Implies that 2^{n} - 1 is a composite number.