Congruence of Product

Theorem
Let $$a, b \in \Z$$, and $$n \in \N$$ such that $$n \ne 0$$.

Let $$a$$ be congruent to $b$ modulo $n$, i.e. $$a \equiv b \pmod n$$.

Then $$\forall m \in \Z$$:
 * $$m a \equiv m b \pmod n$$;
 * $$m > 0 \implies m a \equiv m b \pmod {m n}$$;

Proof

 * From Congruence (Number Theory) is an Equivalence, $$m \equiv m \pmod n$$.

Thus from the definition of Modulo Multiplication, it follows that $$m a \equiv m b \pmod n$$.


 * By definition of congruence modulo $n$:

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