Here we need to just find one counter example that contradicts the validity of the given statement.

Suppose we set values a = 2, b=2, c= 4, d=9, and m=5 in

\(\displaystyle{a}\equiv{b}\pm{o}{d}{m}\)

We get,

\(\displaystyle{2}\equiv{2}\pm{o}{d}{5}\), thus this congruence is satisfied.

Next let us plug these values in \(\displaystyle{c}\equiv{d}\pm{o}{d}{m}\)

We get,

\(\displaystyle{4}\equiv{9}\pm{o}{d}{5}\), observe that \(\displaystyle{9}={4}+{1}\times{5}\), thus

\(\displaystyle{4}\equiv{4}\pm{o}{d}{5}\), again this congruence is also satisfied.

Lastly we plug these values into \(\displaystyle{a}^{{{c}}}\equiv{b}^{{{d}}}\pm{o}{d}{m}\)

We get,

\(\displaystyle{2}^{{{4}}}\equiv{2}^{{{9}}}\pm{o}{d}{5}\)

\(\displaystyle{16}\equiv{512}\pm{o}{d}{5}\), observe that

\(\displaystyle{16}={1}+{3}\times{5}\) and \(\displaystyle{512}={2}+{102}\times{5}\), thus

\(\displaystyle{1}\pm{o}{d}{5}\equiv{2}\pm{o}{d}{5}\), which is incorrect

Thus a counterexample could be the set of values

a=2, b=2, c=4, d=9, and m =5

Suppose we set values a = 2, b=2, c= 4, d=9, and m=5 in

\(\displaystyle{a}\equiv{b}\pm{o}{d}{m}\)

We get,

\(\displaystyle{2}\equiv{2}\pm{o}{d}{5}\), thus this congruence is satisfied.

Next let us plug these values in \(\displaystyle{c}\equiv{d}\pm{o}{d}{m}\)

We get,

\(\displaystyle{4}\equiv{9}\pm{o}{d}{5}\), observe that \(\displaystyle{9}={4}+{1}\times{5}\), thus

\(\displaystyle{4}\equiv{4}\pm{o}{d}{5}\), again this congruence is also satisfied.

Lastly we plug these values into \(\displaystyle{a}^{{{c}}}\equiv{b}^{{{d}}}\pm{o}{d}{m}\)

We get,

\(\displaystyle{2}^{{{4}}}\equiv{2}^{{{9}}}\pm{o}{d}{5}\)

\(\displaystyle{16}\equiv{512}\pm{o}{d}{5}\), observe that

\(\displaystyle{16}={1}+{3}\times{5}\) and \(\displaystyle{512}={2}+{102}\times{5}\), thus

\(\displaystyle{1}\pm{o}{d}{5}\equiv{2}\pm{o}{d}{5}\), which is incorrect

Thus a counterexample could be the set of values

a=2, b=2, c=4, d=9, and m =5