proof writing - Proving divisibility of expression using induction

Question:

Part a:

Prove that for any $b\in \Bbb N,$ if $2^{3b -1} + 5 . 3^{b}$ is divisible by $11$, then $2^{3(b+2) -1} + 5 . 3^{b+2}$ is divisible by $11$.

Part b:

Is statement 1 or statement 2 true? Explain answer.

1. For any odd number $a\in \Bbb N$, $2^{3a -1} + 5 . 3^{a}$ is divisible by $11$




2. For any even number $a\in \Bbb N$, $2^{3a -1} + 5 . 3^{a}$ is divisible by $11$

My attempt:

Part a:

I am not sure what the base case should be.

Induction hypothesis: Assume $2^{3k -1} + 5 . 3^{k}$ is divisible by $11$, for some $k$ natural number.

I am not sure how to prove true for $2^{3(k+2) -1} + 5 . 3^{k+2}$.

Part b:

Would statement 2 be correct since the expression is divisible by $11$ when $a=2$