Linear congruence

A Linear Congruence is a congruence mod p of the form

$ax+b\equiv c\pmod{p}$

where $a, b, c$ , and $p$ are constants and $x$ is the variable to be solved for.

Example I: How to solve

Say $5x\equiv 7\pmod{8}$ . Find $x$ .

Solution

$5x\equiv 7\equiv 15\pmod{8}$ , so

$x \equiv 3 \pmod 8$ . Note that we can divide by 5 because 5 and 8 are relatively prime.

An alternative method is to note that $5\cdot 5 \equiv 25 \equiv 1 \pmod{8}$ , so

$5\cdot5x\equiv 5\cdot7\pmod{8}$ and thus

$x \equiv 3 \pmod 8$ .

Note that not every linear congruence has a solution. For instance, the congruence equation $2x \equiv 3 \pmod 8$ has no solutions. A solution is guaranteed if and only if $a$ is relatively prime to $p$ . If $a$ and $p$ are not relatively prime, say with greatest common divisor $d$ , then we have two options:

if $d$ divides $b$ , there will be a solution $\pmod \frac{p}{d}$
if $d$ does not divide $b$ , there will be no solution.

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Linear congruence

Example I: How to solve

Solution