Z1) Find all integer values that the expression

$$
\frac{p q+p^{p}+q^{q}}{p+q}
$$

where $p$ and $q$ are prime numbers.

Answer: The only integer value is 3 .
Solution: If $p$ and $q$ are both odd, then the numerator is odd and the denominator is even. Because an even number never divides an odd number, this does not yield an integer value, so we can assume that at least one of our primes is even, and therefore equal to 2. Because the expression is symmetric in $p$ and $q$, we can assume without restriction of generality that $q=2$.

So we enter $q=2$ and it is now sufficient to enter all integer values of the expression

$$
\frac{2 p+p^{p}+4}{p+2}=\frac{2(p+2)+p^{p}}{p+2}=2+\frac{p^{p}}{p+2}
$$

to find.

For this to be an integer, $p+2$ must be a divisor of $p^{p}$. But because $p$ is a prime number, all positive divisors of $p^{p}$ are the following: $1, p, p^{2}, \ldots, p^{p-1}$ and $p^{p}$. If $p>2$, then we have

$$
p2$ the number $p+2$ never divides $p^{p}$ and we do not get an integer value. The only case that remains is $p=q=2$, which results in a value of 3.