Question

# Find all whole number solutions of the congruence equation. (2x + 1) -= 5 mod 4

Congruence
Find all whole number solutions of the congruence equation.
$$\displaystyle{\left({2}{x}+{1}\right)}\equiv{5}\text{mod}{4}$$

2021-02-13
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,...