\section*{Problem 2}

Find the smallest positive integer which can be represented as \(36^{\mathrm{m}}-5^{\mathrm{n}}\).

\section*{Answer}

11


\section*{Solution}

Obviously \(11=36^{1}-5^{2}\), and we guess that this is the best possible. We cannot have \(36^{\mathrm{m}}-5^{\mathrm{n}}=\) \(\mathrm{k}\), where 2,3 or 5 divides \(\mathrm{k}\) (because then 2 would divide \(5^{\mathrm{n}}\) and similarly in the other cases). So the only possible values of \(\mathrm{k}<11\) are 1,7 .

We have \(36^{m}-5^{n}=1 \bmod 5\), so \(k \neq 7\). Similarly, \(36^{m}-5^{n}=3 \bmod 4\), so \(k \neq 1\).

