# Math Help - Compute 3^80 (mod 7)

1. ## Compute 3^80 (mod 7)

I cannot remember how to do these problems and they are due right after break!!!
Compute 3 ^80 (mod7) and find all integers x such that 5x is congruent to 1 (mod 100)

2. #1: Note that: $3^6 \equiv 1 \ (\text{mod } 7)$

So: \begin{aligned} \left(3^6\right)^{{\color{red} 13}} & \equiv 1^{{\color{red} 13}} & (\text{mod } 7) \\ 3^{78} & \equiv 1 & (\text{mod } 7) \end{aligned}

Multiply both sides by the appropriate power of 3 and you'll get your result.

#2: If $(a,m) \! \not{\mid} \ b$, then $ax \equiv b \ (\text{mod }n)$ has no solutions.