# Cancellability of Congruences/Corollary 1/Proof 3

Jump to navigation
Jump to search

## Corollary to Cancellability of Congruences

Let $c$ and $n$ be coprime integers, that is:

- $c \perp n$

Then:

- $c a \equiv c b \pmod n \implies a \equiv b \pmod n$

## Proof

Let:

- $c a \equiv c b \pmod n$

Then by definition of congruence:

- $n \divides k \paren {x - y}$

We have that:

- $c \perp n$

Thus by Euclid's Lemma:

- $n \divides \paren {x - y}$

So by definition of congruence:

- $a \equiv b \pmod n$

$\blacksquare$

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 6$: Examples of Finite Groups: $\text{(iii)} \ \text{(D)}$