Prove by induction that 7^(8n+3) + 2 is divisible by 5, where n is a natural number.

We need to first consider the base case, n=0. (We consider the smallest possible value of n as the base case.)7^(80+3) + 2 = 343 + 2 = 345 = 69 * 5, so the statement is true for n=0.
Assume true for n = k, i.e. 7^(8k+3) +2 is divisible by 5.
Inductive step: Consider n = k+1. (To prove by induction we need to show that the statement is true for n = k+1)We know that 7^(8k+3) +2 is divisible by 5, and this fact needs to be used in the inductive step. Start by replacing n with k+1:7^(8(k+1)+3) + 2= 7^(8k +11) + 2 = 7^8 * (7^(8k+3)+2) - 2(7^8) +2 (by laws of indices)Here is when we use the fact that 7^(8k+3) +2 is divisible by 5. We do this by replacing 7^(8k+3) +2 with 5a, where a is some integer.= 7^8 * 5a - 11529600 (11529600 is 2*(7^8) - 2 -> this can be done using a calculator.)= 5 * (7^8 - 2305920)
We end with the inductive statement:The statement is true for n=0, and if true for n=k then true for n=k+1, and so, by induction, the statement is true for all n, where n is a natural number.