Proving divisibility by using induction: $133 \mid (11^{n+2} + 12^{2n+1})$

Question

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

Answer 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.