# A Counterexample to the statement, If ac\equiv bc\pmod m, where a,b,c, and m are integers with m \geq 2, then a\equiv b \pmod m.

Congruence

A Counterexample to the statement, If $$ac\equiv bc\pmod m$$, where a,b,c, and m are integers with $$\displaystyle{m}\geq{2}$$, then $$a\equiv b \pmod m$$.

2021-05-02

Here we need to just find one counter example that contradicts the validity of the given statement.
Suppose we set values a=0, b=1, c=3, m=4 in $$ac\equiv bc\pmod{m}$$
We get,
$$0\times 1\equiv 1\times 3\pmod 3$$
$$0=3\pmod 3$$
0=0
Thus, this congruence is satisfied.
Next let us plug these values in $$a\equiv b\pmod m$$
We get,
$$0\equiv 1\pmod 3$$, which is obviously incorrect.
Thus a counterexample could be the set of value
a=0, b=1, c=3, m=4