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)
#1: Note that: $\displaystyle 3^6 \equiv 1 \ (\text{mod } 7)$
So: $\displaystyle \begin{aligned} \left(3^6\right)^{{\color{red} 13}} & \equiv 1^{{\color{red} 13}} & (\text{mod } 7) \\ 3^{78} & \equiv 1 & (\text{mod } 7) \end{aligned}$
Multiply both sides by the appropriate power of 3 and you'll get your result.
#2: If $\displaystyle (a,m) \! \not{\mid} \ b$, then $\displaystyle ax \equiv b \ (\text{mod }n)$ has no solutions.