To solve the congruence \(\displaystyle{5}+{x}\equiv{2}{\left({b}\text{mod}{6}\right)}\), try each of the numbers

0, 1,2,3,4, and 5 to see which one solves the congruence.

For x=0

\(\displaystyle{5}+{0}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{0}\equiv{5}{\left({b}\text{mod}{6}\right)}\). So,

x = 0 is not the solution of the given congruence.

For x=1

\(\displaystyle{5}+{1}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{1}\equiv{0}{\left({b}\text{mod}{6}\right)}\). So,

x= 1 is not the solution of the given congruence.

For x =2

\(\displaystyle{5}+{2}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{2}\equiv{1}{\left({b}\text{mod}{6}\right)}\). So,

x = 2 is not the solution of the given congruence.

For x=3

\(\displaystyle{5}+{3}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{8}\equiv{2}{\left({b}\text{mod}{6}\right)}\), and it is true. So, x = 3 is the solution of the given congruence.

For x=4

\(\displaystyle{5}+{4}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{9}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{9}\equiv{3}{\left({b}\text{mod}{6}\right)}\). So, x =4

is not the solution of the given congruence.

For x=5

\(\displaystyle{5}+{5}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{10}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{10}\equiv{4}{\left({b}\text{mod}{6}\right)}\). So, x = 5

is not the solution of the given congruence.

Thus, x = 3 is the solution of the given congruence

\(\displaystyle{5}+{x}\equiv{2}{\left({b}\text{mod}{6}\right)}\).

Final Statement:

Therefore, the solution of the congruence \(\displaystyle{5}={x}\equiv{2}{\left({b}\text{mod}{6}\right)}\) is x=3.

0, 1,2,3,4, and 5 to see which one solves the congruence.

For x=0

\(\displaystyle{5}+{0}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{0}\equiv{5}{\left({b}\text{mod}{6}\right)}\). So,

x = 0 is not the solution of the given congruence.

For x=1

\(\displaystyle{5}+{1}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{1}\equiv{0}{\left({b}\text{mod}{6}\right)}\). So,

x= 1 is not the solution of the given congruence.

For x =2

\(\displaystyle{5}+{2}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{5}+{2}\equiv{1}{\left({b}\text{mod}{6}\right)}\). So,

x = 2 is not the solution of the given congruence.

For x=3

\(\displaystyle{5}+{3}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{8}\equiv{2}{\left({b}\text{mod}{6}\right)}\), and it is true. So, x = 3 is the solution of the given congruence.

For x=4

\(\displaystyle{5}+{4}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{9}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{9}\equiv{3}{\left({b}\text{mod}{6}\right)}\). So, x =4

is not the solution of the given congruence.

For x=5

\(\displaystyle{5}+{5}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

\(\displaystyle\Rightarrow{10}\equiv{2}{\left({b}\text{mod}{6}\right)}\), but this is not true since \(\displaystyle{10}\equiv{4}{\left({b}\text{mod}{6}\right)}\). So, x = 5

is not the solution of the given congruence.

Thus, x = 3 is the solution of the given congruence

\(\displaystyle{5}+{x}\equiv{2}{\left({b}\text{mod}{6}\right)}\).

Final Statement:

Therefore, the solution of the congruence \(\displaystyle{5}={x}\equiv{2}{\left({b}\text{mod}{6}\right)}\) is x=3.