Exercise 7

Find counterexamples to the following statements:

1. If $ac \equiv bc \pmod m$ where $a, b, c, m \in \Z$ with $m \ge 2$, then $a \equiv b \pmod m$.
2. If $a \equiv b \pmod m$ and $c \equiv d \pmod m$ where $a, b, c, m \in \Z$ with $m \ge 2$, then $a^c \equiv b^d \pmod m$.

The first part of this exercise should convince you: you cannot “divide” both sides by $c$ in modular arithmetic!