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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
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asked 2021-04-30

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 \(a\equiv b \pmod m\).

Answers (1)

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

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