<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] Hint:
Let $f(n)=5^{5n+1}+4^{5n+2}+3^{5n}$. Then
$$
f(n+1)-f(n)
=(5^5-1) \cdot5^{5 n + 1} + (4^5-1) \cdot 4^{5 n + 2} + (3^5-1) \cdot 3^{5n}
= 11(284\cdot5^{5 n + 1}+ 93 \cdot 4^{5 n + 2} +22\cdot 3^{5n})
$$
The crucial point is that $5^5-1,4^5-1,3^5-1$ are all divisible by $11$.
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