Step 1

Consider the given congruence equation.

\(\displaystyle{\left({2}{x}+{1}\right)}\equiv{5}\text{mod}{4}\)

Step 2

Substitute each whole number less than 4 into the congruence equation.

\(\displaystyle{x}={0},{2}{\left({0}\right)}+{1}\equiv{5}\text{mod}{4}\) a solution

\(\displaystyle{x}={1},{2}{\left({1}\right)}+{1}\ne{5}\text{mod}{4}\) not a solution

\(\displaystyle{x}={2},{2}{\left({2}\right)}+{1}\equiv{5}\text{mod}{4}\) a solution

\(\displaystyle{x}={3},{2}{\left({3}\right)}+{1}\ne{5}\text{mod}{4}\) not a solution

The solution between 0 and 3 is 0 and 2.

The remaining solutions are determined by repeatedly adding the modulus, 4, to these solutions.

Hence, the solutions to the congruence equation 0,2,4,6,8,10,...

Consider the given congruence equation.

\(\displaystyle{\left({2}{x}+{1}\right)}\equiv{5}\text{mod}{4}\)

Step 2

Substitute each whole number less than 4 into the congruence equation.

\(\displaystyle{x}={0},{2}{\left({0}\right)}+{1}\equiv{5}\text{mod}{4}\) a solution

\(\displaystyle{x}={1},{2}{\left({1}\right)}+{1}\ne{5}\text{mod}{4}\) not a solution

\(\displaystyle{x}={2},{2}{\left({2}\right)}+{1}\equiv{5}\text{mod}{4}\) a solution

\(\displaystyle{x}={3},{2}{\left({3}\right)}+{1}\ne{5}\text{mod}{4}\) not a solution

The solution between 0 and 3 is 0 and 2.

The remaining solutions are determined by repeatedly adding the modulus, 4, to these solutions.

Hence, the solutions to the congruence equation 0,2,4,6,8,10,...