# Linear congruence

A **Linear Congruence** is a congruence mod p of the form
where , , , and are constants and is the variable to be solved for.

## Solving

Note that not every linear congruence has a solution. For instance, the congruence equation has no solutions. A solution is guaranteed iff is relatively prime to . If and are not relatively prime, let their greatest common divisor be ; then:

- if divides , there will be a solution
- if does not divide , there will be no solution

## Example

### Problem

Given , find .

### Solution 1

, so . Thus, . Note that we can divide by because and are relatively prime.

### Solution 2

Multiply both sides of the congruence by to get . Since and , .