Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).
Katie and Charlotte had \(4\) sheets of paper. They decided to cut some of the sheets into \(4\) pieces, then, some of the newly obtained papersheets they also cut into \(4\). In the end they counted the number of all sheets. Could this number be \(2024\)?