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Given a linear congruence $x + a ≡ y $ mod $n$, is it valid to rewrite as $x - y ≡ a $ mod $n$ ? Based on the definition of congruence, $n | x + a - y$ if we have the expression $x + a ≡ y $ mod $n$ and $n | x - y - a$ if we have the expression $x - y ≡ a $ mod $n$

Bill Dubuque
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errorcodemonkey
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    Yes, this follows immediately from the linked Congruence Sum Rule, i.e. add $-a-y$ to both sides of the congruence (or subtract $a$ then $y).,$ Conceptually it is better to learn these arithmetical laws which allows us to manipulate congruences just as we do equations (vs. rewrite them as divisibility relations). – Bill Dubuque Aug 24 '20 at 16:37

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What you are looking for is

$$x-y\equiv -a(\text{mod }n)$$

You can perform addition and subtraction using modular arithmetic just as you would like, but the signs remain important.

Josh B.
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