Find The Remainder

We note that $3^2 \equiv 1 \pmod 8$ , therefore, we have:

$\begin{aligned} 3^1 + 3^3 + 3^5+...+3^{29} & \equiv 3(1 + 3^2 + 3^4+...+3^{28}) \\ & \equiv 3(1+1+1 + ...+1) \pmod 8 \\ & \equiv 3(15) \pmod 8 \\ & \equiv \boxed{5} \pmod 8 \end{aligned}$

We want to see the remainder when using $(mod 8)$ .

$3^{1}\equiv 3$ $(mod 8)$

$3^{2}\equiv 1$ $(mod 8)$

$3^{3}\equiv 3$ $(mod 8)$

$3^{5}\equiv 3$ $(mod 8)$

We see that for any odd power for the number $3$ is equivalent to $3$ $(mod 8)$ .

Adding the first fifteen odd power for the number $3$ , and using equivalent properties :

$3^{1}+3^{3}+3^{5}+...+3^{29} \equiv 3 \times 15 (mod 8) \equiv 5(mod 8)$

So the remainder is $5$ .