Solution of Linear Congruence
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Theorem
Let $a x \equiv b \pmod n$ be a linear congruence.
The following results hold:
Existence
$a x \equiv b \pmod n$ has at least one solution if and only if:
- $\gcd \set {a, n} \divides b$
that is, if and only if $\gcd \set {a, n}$ is a divisor of $b$.
Condition for Uniqueness
$a x \equiv b \pmod n$ has a unique solution if and only if $\gcd \set {a, n} = 1$.
Number of Solutions
Let $\gcd \set {a, n} = d$.
Then $a x \equiv b \pmod n$ has $d$ solutions which are given by the unique solution modulo $\dfrac n d$ of the congruence:
- $\dfrac a d x \equiv \dfrac b d \paren {\bmod \dfrac n d}$
Examples
Solution to $5 x = 4 \pmod 3$
Let $5 x = 4 \pmod 3$.
Then:
- $x = 2 + 3 u$
where $u \in \Z$.
Solution to $7 x = 6 \pmod 5$
Let $7 x = 6 \pmod 5$.
Then:
- $x = 3 + 5 u$
where $u \in \Z$.
Solution to $9 x = 8 \pmod 7$
Let $9 x = 8 \pmod 7$.
Then:
- $x = 4 + 7 t$
where $t \in \Z$.