How to calculate modulo negative number?

Question

Why 134 mod -26 is equal 22?

I found an explanation, which says: 134 = (-26 * -6) - 22

However, 134 = (-26*-5)+4, so why 134 mod -26 is not equal 4

if 134=(26*5)+4 => 134 mod 26=4

All i could find, is about a situation, if i take a modulo of negative number (-134 mod 26), but that is not what i want.

$a\equiv b\bmod n$ by definition is $n\mid (b-a)$. Check this for $a=134$ and $b=22$ and $n=-26$. So it is not true. — Dietrich Burde, Nov 01 '21 at 19:16

Ethan Bolker · Accepted Answer · 2021-11-01T19:41:33.377

The first sentence in the post is wrong. $-136$ in not congruent to $22$ modulo $-26$. It is congruent to $-22$.

In the second sentence you had an error (now fixed). You mean $-26 \times -5$.

$4$ and $-22$ are the same answer modulo $-26$ since they differ by a multiple of $-26$.

Arithmetic modulo $m$ is essentially the same as arithmetic modulo $-m$.

Sometimes you want to require an answer that lies between $0$ and the positive value $|m|-1$, sometimes any equivalent answer will do.

If you are working with or writing a computer program to do modular arithmetic the program will make a choice - usually the representative between $0$ and $|m|-1$.

How to calculate modulo negative number?

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