# For the following statement, either prove that they are true or provide a counterexample:Let a, b, c, m in Z such that m > 1. If ac -= bc (mod m), then a -= b (mod m)

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, $m\in Z$ such that m > 1. If

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Let a, b, c, $m\in Z$ such that m > 1. If
This statement is false.
Counterexample:
$4\mid 4$
$4\mid 16-12$
By using definition of congruence,
$12\equiv 16\phantom{\rule{1em}{0ex}}\mathrm{mod}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}\left(4\right)$
Now,
$12=6\cdot 2$ and $16=8\cdot 2$
$6\cdot 2\equiv 8\cdot 2\phantom{\rule{1em}{0ex}}\mathrm{mod}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}\left(4\right)$
Compare it with $ac\equiv bc\left(\phantom{\rule{1em}{0ex}}\mathrm{mod}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}m\right)$,
We get c = 2.
But $6!\equiv 8\phantom{\rule{1em}{0ex}}\mathrm{mod}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}\left(4\right)$
as $4!\mid 8-6$
Therefore, If is false.