<s>[INST] Show that $5^n\mid2^{5^{n-1}}+3^{5^{n-1}}$ for $n \gt 0$. Show that $5^n\mid 2^{5^{n-1}}+3^{5^{n-1}}$ for $n \gt 0$.
I tried using Euler's Theorem: $2^{4\cdot5^{n-1}} \equiv 1 \mod{5^n}$ and $3^{4\cdot5^{n-1}}  \equiv 1 \mod{5^n}$ but I can't find a way to use it.
I also tried showing that the sum of the remainders of $2^{5^{n-1}}+3^{5^{n-1}}$ is a multiple of $5^n$ with induction but got stuck again.
How do I prove this kind of statement?
 [/INST] Hint:
If $a+b=c5^r$ where $a,b,c$ are integers
$$(a+b)^5=a^5+b^5+\binom51ab(a^3+b^3)+\binom52a^2b^2(a+b)$$
$$\implies a^5+b^5=(a+b)^5-5ab(a^3+b^3)-10a^2b^2(a+b)$$ is clearly divisible by $5^{r+1}$
Set $a=2,b=5d-2,$ where integer $d$ is not divisible by $5$  then $r=1$
By induction, we can establish the proposition
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