Question

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

Polynomial factorization
ANSWERED
asked 2021-04-22
To solve:
The congruence \(\displaystyle{5}+{x}\equiv{2}{\left({b}\text{mod}{6}\right)}\)

Answers (1)

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.
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