Congruent Modulo with negative

Question

Given $16 \equiv 7 \pmod m$.

Find $m$:

$$7-16 = -9$$

so, the $m$ can be what ever can divide into $-9$ ?

The answer is yes if you meant that m can be anything that divides m... — DonAntonio, Jun 04 '13 at 21:54
So If I have the following options a] m = mod 2, b] m = mod 5, c] m = mod 7, d] m = mod 10, e] mod 3 :: the answer is mod 3? [e]? — MethodManX, Jun 04 '13 at 21:57
Yes @MethodManX...but again: what's your definition of "equality modulo m"? — DonAntonio, Jun 04 '13 at 22:00
sigh...ok, @MethodManX: so how do you define "a congruent to b modulo m", or $,a=b\pmod m,$ ? — DonAntonio, Jun 04 '13 at 22:38

score 1 · Answer 1 · answered Jun 04 '13 at 21:53

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$$16=7\pmod m\iff 16-7=9=0\pmod m\iff m\mid 9\implies m=\ldots$$

answered Jun 04 '13 at 21:53

DonAntonio

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I can't understand what you are saying @ DonAntonio – MethodManX Jun 04 '13 at 21:55
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What's the definition of $,a=b\pmod m,$ you know, @MethodManX ? – DonAntonio Jun 04 '13 at 21:56

score 0 · Answer 2 · answered Jun 04 '13 at 22:17

Note that: $$16 \equiv 7 \mod m$$ means $$16=7+km,\text{ for some } k\in \mathbb Z$$ And if we put $r=-k$ we can formulate the alternative $$7=16-km=16+rm\text{ with }r\in\mathbb Z$$

So we have either $9=km,$ or $-9=-km=rm$. The two formulations are entirely equivalent. $m$ is taken as being positive.

Congruent Modulo with negative

2 Answers2