<s>[INST] Proving that $19\mid 5^{2n+1}+3^{n+2} \cdot 2^{n-1}$ How can I prove that $$5^{2n+1}+3^{n+2} \cdot 2^{n-1} $$ can be divided by 19 for any nonnegative n? What modulo should I choose?
 [/INST] $2^{n-1} = \frac12 2^n \equiv 10 \cdot 2^n \pmod {19}$. Hence:
$5^{2n+1} +3^{n+2} \cdot 2^{n-1} \equiv 5 \cdot 6^n + 90 \cdot 6^n \pmod {19} \equiv 95 \cdot 6^n \pmod {19} \equiv 0 \pmod {19}$
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