Old Numbert Th Problem No1

[rsoldo]

For which the natural numbers $n$ number $5^n+n^5$ is divisible by 13? Which is the smallest such $n$?

[zykov]

$5^n \mod 13$ has period 4 and values $(5, 12, 8, 1)$.
$n^5 \mod 13$ has period 13 and values $(1, 6, 9, 10, 5, 2, 11, 8, 3, 4, 7, 12, 0)$.

[zykov]

In $(5, 12, 8, 1)$:
1:5 matches 8:8 - $n=21+52k$, smallest $n=21$
2:12 matches 1:1 - $n=14+52k$, smallest $n=14$
3:8 matches 5:5 - $n=31+52k$, smallest $n=31$
4:1 matches 12:12 - $n=12+52k$, smallest $n=12$

The smallest $n=12$.