Proving divisibility by using induction: $133 \mid (11^{n+2} + 12^{2n+1})$ If $n > 0$, then prove the following by using induction: $$133|(11^{n+2} + 12^{2n+1}).$$

Inductive step:
$$\begin{align}11^{(n+1) + 2} + 12^{2(n+1)+1} &= 11^{n+3} + 12^{2n+3}\\
&=11\cdot11^{n+2} + 144\cdot12^{2n+1}\\
&= 11\cdot11^{n+2} + 11\cdot12^{2n+1} + 133\cdot12^{2n+1}\\
&=11\cdot(11^{n+2} + 12^{2n+1}) + 133\cdot12^{2n+1}\end{align}$$
But $133 | (11^{n+2} + 12^{2n+1})$. This is the inductive hypothesis. Hence $133$ must divide the above. 
