<s>[INST] Divisibility of an expression by 11 How to prove that $5^{5n+1}+4^{5n+2}+3^{5n}$ where $n\in \mathbb{N}$ is divisible by $11$ using mathematical induction?
I have tried and got to this $$5 \cdot 25 \cdot 25 \cdot 5^{5k+1}+4\cdot 16 \cdot 16 \cdot 4^{5k+2}+3\cdot 9 \cdot 9 \cdot 3^{5k}$$
and I even skipped the first step.
Thanks in advance.
(I am sorry if the tags are not relevant to this question)
 [/INST] Let $f(n)=5^{5n+1}+4^{5n+2}+3^{5n}$
$$f(m+1)-3^5\cdot f(m)=5^{5(m+1)+1}+4^{5(m+1)+2}-3^5(5^{5m+1}+4^{5m+2})$$
$$=5^{5m+1}(5^5-3^5)+4^{5m+2}(4^5-3^5)$$  which is divisible by $11$
as  $3^5=9(22+5)\equiv1\pmod{11}, 5^5=5^2\cdot5^3\equiv3\cdot4\equiv1\pmod{11}$
Similarly, $4^5\equiv1\pmod{11}$
$$\implies11|f(m)\iff11\mid f(m+1)$$
</s>