Prove $10^{n+1}+3\cdot 10^n+5$ is divisible by $9$? How do I prove that an integer of the form $10^{n+1}+3\cdot 10^{n}+5$ is divisible by $9$  for $n\geq 1$?I tried proving it by induction and could prove it for the Base case n=1. But  got stuck while proving the general case. Any help on this ? Thanks. 

I would normally use congruences, but this can be done explicitly!
$$
10^{n+1}+3\cdot 10^{n}+5
= 9 \cdot \underbrace{11 \cdots 1}_{n+1} + 1
+
9 \cdot \underbrace{33 \cdots 3}_{n} + 3
+ 5
=\\=
9 \cdot (\underbrace{11 \cdots 1}_{n+1} + \underbrace{33 \cdots 3}_{n} + 1) = 9 \cdot 1\underbrace{44\cdots 4}_{n-1}5.
$$
