Ta có:
$11^{n+2}+12^{2n+1}=121.11^n+12.144^n\equiv 2.4^n+5.4^n\;($mod $7) $$\Rightarrow 11^{n+2}+12^{2n+1}\equiv 7.4^n\equiv0\;($mod $7)$
$11^{n+2}+12^{2n+1}=121.11^n+12.144^n\equiv 7.11^n+5.11^n\;($mod $19) $$\Rightarrow 11^{n+2}+12^{2n+1}\equiv 19.11^n\equiv0\;($mod $19)$
Mà $gcd(7;19)=1 \Rightarrow 11^{n+2}+12^{2n+1}\equiv0\;($mod $133),\forall n\in\mathbb{N}$