A composite Mersenne number

Statement

Mersenne numbers have the form $M_p=2^p-1$. Even when $p$ is prime, $M_p$ need not be prime. Show that $M_{11}=2^{11}-1=2047$ is composite by finding its smallest prime divisor.

Solution

$2047=2^{11}-1=23\cdot 89.$ The smallest prime divisor is $\boxed{23}$. So $M_{11}$ is composite, even though $11$ is prime.