Find the valuation...

(i) 3-adic of $A=2^{27}+1$.

(ii) 7-adic of $B=161^{14}-112^{14}$.

(iii) 2-adic of $C=7^{20}+1$.

(iv) 2-adic of $D=17^{48}-5^{48}$.
(i) We can directly apply LTE: $v_{3}(A)=v_{3}(2+1)+v_{3}(27)=4$.

(ii) Be careful, $161=7 \cdot 23$ and $112=7 \cdot 16$, so we must first extract the factors of 7 before applying LTE: $v_{7}(B)=v_{7}\left(7^{14}\right)+v_{7}\left(23^{14}-16^{14}\right)=14+v_{7}(23-16)+v_{7}(14)=16$.

(iii) Argh, LTE does not cover this case. But in fact, we observe that $C \equiv 2$ modulo 4, so the valuation is 1.

(iv) We apply LTE: $v_{2}(D)=v_{2}(17-5)+v_{2}(17+5)+v_{2}(48)-1=2+1+4-1=6$.