Partition of a prime $p$ into primes raised to a power to be dvisible by $p$ Find all the partitions of prime $p$ in primes less than $p$. Raise each term in each partition by some power $k > 1$ to see if the sum of these terms will be divisible by $p$. Of course, $k$ can differ for each partition. For example, for $5$, $2^3 + 3^3 = 35 = 5 \times 7$ and for $11$, $4 \times 2^4 + 3^4 = 209 = 11 \times 19$. Do you think all partitions will eventually have  some least power $k$? Will it be less than $p$ itself?

There always exists such a least power $k > 1$, and it will certainly never be greater than $p$, because $p$ itself is a valid power:
By Fermat's little theorem, $a^p \equiv a \bmod p$. So if $a_1 + \ldots + a_n$ is divisible by $p$, then so is ${a_1}^p + \ldots + {a_n}^p$.
