Problem 2. a) Simplify the fraction $\frac{64}{\overline{a b 64}-36 \cdot \overline{a b}}, a \neq 0$, so that it becomes irreducible.

b) Prove that the number $n=3^{23}+5^{23}+15^{23}$ is divisible by 23.
## Solution and Scoring

a) $\overline{a b 64}-36 \cdot \overline{a b}=100 \cdot \overline{a b}+64-36 \cdot \overline{a b}=64 \cdot(\overline{a b}+1)$; after simplification we get: $\frac{1}{\overline{a b}+1} \ldots \ldots \ldots \ldots \ldots . .3$ points

b) $n=3^{23}+5^{23}+15^{23}=3 \cdot 3^{22}+5 \cdot 5^{22}+15 \cdot 15^{22}=$.............................................................. 2 points

$=3 \cdot\left(M_{23}+1\right)+5 \cdot\left(M_{23}+1\right)+15 \cdot\left(M_{23}+1\right)=M_{23}+23=M_{23} \quad$................................................. 2 points