a) $11| (3^n+4^n)$ $11|(9^n+12^n)$ $11|9^n+1$, but $9^n+1 \equiv 10,5,4,6,2$ mod $11$, so there are not solutions. b)$31|(4^n+7^n+20^n)$ $31|(32^n+56^n+160^n)$ $31|(1+25^n+5^n)$ $5^{3m}\equiv 1$ mod $31$ $5^{3m+1}\equiv 5$ mod $31$ $5^{3m+2}\equiv 25$ mod $31$ $25^{3m}\equiv 1$ mod $31$ $25^{3m+1}\equiv 25$ mod $31$ $25^{3m+2}\equiv 5$ mod $31$ So $1+5^n+25^n \equiv 3,0,0$ mod 31 $n=3k+1,3k+2$