Integer Coprime to Modulus iff Linear Congruence to 1 exists/Corollary
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Let $p$ be a prime number.
The linear congruence:
- $a x \equiv 1 \pmod p$
has a solution $x$ if and only if $a \not \equiv 0 \pmod p$.
By definition of congruence:
- $a \not \equiv 0 \pmod p \iff p \nmid a$
where $p \nmid a$ denotes that $p$ is not a divisor of $a$.
From Prime not Divisor implies Coprime:
- $p \nmid a \iff p \perp a$
where $p \perp a$ denotes that $p$ and $a$ are coprime.
The result follows from Integer Coprime to Modulus iff Linear Congruence to 1 exists.
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences: Proposition $2$: Corollary