Mathematics B (MEI) H630: PROOF Topical Past Paper Questions

Prove that, when $n$ is an even number, $n^{3} + 4$ is a multiple of 4 but not a multiple of 8. [3]

A student makes the following conjecture. For all positive integers n, $6n-1$ is always prime. For all positive integers n, $ 6n-1 $ is always prime. Use a counter example to disprove this conjecture.

(a) Show that $4! < 4^{4}$. [2] (b) Nina believes that the statement $n! < n^{n}$ is true for all positive integers $n$. Prove that Nina is not correct. [2]