Show that for each \(n=1\), \(2\), \(3\), . . ., we have \(n<2^n\).
Show that \(n^2+n+1\) is not divisible by \(5\) for any natural number \(n\).
Given a natural number \(n\), find a formula for the number of \(k\) less than \(n\) such that \(k\) is coprime to \(n\). Prove that the formula works.
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).