Oxford University MAT 2016 Solved Papers

We are trying to construct counter-examples for each of the statements. Note that $0\le \mathit{x}\left(\mathit{n}\right)\le 9$. (a) is true since, for example, $\mathit{\Pi }\left(4\right)=1$, but 4 is not prime. (b) is false - we don't need to consider even $\mathit{n}$ beyond ${\mathit{x}}_{\mathit{n}}=4$; for this case we know no primes end in a 4 , but for example $\mathit{\Pi }\left(64\right)=1$ as $64=2^6$. For odd $\mathit{n},\mathit{x}\left(\mathit{n}\right)=1,\mathit{\Pi }\left(\mathit{n}\right)=1$, counterexample $\mathit{n}=121={11}^2;\mathit{x}\left(\mathit{n}\right)=3,\mathit{\Pi }\left(\mathit{n}\right)=1$, counterexample $\mathit{n}=243=3^5;\mathit{x}\left(\mathit{n}\right)=5,\mathit{\Pi }\left(\mathit{n}\right)=1$, counterexample $\mathit{n}=25=5^2;\mathit{x}\left(\mathit{n}\right)=7,\mathit{\Pi }\left(\mathit{n}\right)=1$, counterexample $\mathit{n}=16807=7^5;\mathit{x}\left(\mathit{n}\right)=9,\mathit{\Pi }\left(\mathit{n}\right)=1$, counterexample $\mathit{n}=9=3^2$. (c), (d), and (e) are all true. The answer is (b).