Problem

Source: X International Festival of Young Mathematicians Sozopol 2019, Theme for 10-12 grade

Tags: number theory, Divisibility, greatest common divisor, modulo



Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$.