How did we get the result in the 2nd iff statement?

Question

I don't understand the reasoning behind 10 is congruent to one under mod 9 implies that we can get the third statement below :

Can I get an explanation on this?

Answer 1

Since, $10 \equiv 1 \mod 9$

Therefore, $$10^i \equiv1 \mod9$$ Hence, $$a_i 10^i \equiv a_i \mod9$$ for all $i=0,1,2,\cdots n$. Adding these up$$a_0+a_1 10+a_2 10^2 + \cdots a_n 10^n \equiv a_0+a_1 1+a_2 1^2 + \cdots a_n 1^n \mod 9 $$

Answer 2

reduced the $10^n$ to $1^n$ because they have the same remainder on division by 9

Answer 3

since $$10\equiv 1\mod 9$$ we get $$10^n\equiv 1^n\equiv 1\mod 9$$