Let n ∈ N which one of the following is true?

(a) $47^n+16 n-1$$\begin{array}{r} (48-1)^n+4(4 n)-1 \\ 48 k+(-1)^n+4 m-1 \\ 4(12 k+m)+(-1)^n-1 \end{array}$47^n+16 n-1$is divisible by 4 if$n$is odd.$\therefore$It is false for all values of$n$(b)$2\left(4^{2 n+1}\right)-3^{3 n+1}$8 \cdot 4^{2 n}-3 \cdot 3^{14} Put$n=1, \quad 8 \cdot 4^2-3 \cdot 3^1=8 \times 16-9=119$119 is not divisible by 9 . (c)$4^{n+}-3 n-1$Put$n=2, \quad 4^2-3(2)-1=16-7=9$which is not divisible by 11 . (d)$3 \cdot 5^{2 n+1}+2^{3 n+1}\begin{gathered}15 \cdot 5^{2 n}+2 \cdot 2^{3 n} \\ 15 \cdot\left(25^n+2 \cdot(8)^n\right. \\ 15 \cdot(17+8)^n+2 \cdot 8^n \\ 15 \cdot \left(17 k+8^n\right)+2 \cdot 8^n \\ 15\left(17 k+8^n\right)+8^n(15+2) \end{gathered} =17\left(15 k+8^n\right)$\therefore 3 \cdot 5^{3 n+1}+2^{3 n+1}$ is divisible by 17 for all values of $n$