Question

# To solve: The congruence 5+x \equiv 2(\bmod 6)

Polynomial factorization
To solve:
The congruence $$\displaystyle{5}+{x}\equiv{2}{\left({b}\text{mod}{6}\right)}$$

2021-04-24
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.