Division in modular arithmetic

Asked Jun 04 '16 at 13:36

Active Jul 08 '16 at 07:30

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Let $p$ be an odd prime. Given that $a\equiv b \pmod p$ and $c \equiv d \pmod p$, such that none of $a,b,c,d$ is a multiple of $p$. Under what conditions, $\frac{a}{c} \equiv \frac{b}{d} \pmod p$.

asked Jun 04 '16 at 13:36

rationalbeing

1,187

2

HINT:
$$ad-bc=d(a-b)-b(c-d)$$
– lab bhattacharjee Jun 04 '16 at 13:39
You have listed all conditions already. – Servaes Jun 04 '16 at 13:39
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@labbhattacharjee thanks, done! – rationalbeing Jun 04 '16 at 13:43
@Servaes thanks, can you please tell why otherwise this kind of division fails? – rationalbeing Jun 04 '16 at 13:44
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Congruences modulo $p$ only make sense for integers. The notation $\tfrac{1}{c}\pmod{p}$ denotes the residue class of integers that, when multiplied by $c$ are in the residue class of $1$. If $c$ is a multiple of $p$ then no such integer exists, and hence $\tfrac{1}{c}$ is not defined. – Servaes Jun 04 '16 at 13:49
@Servaes thanks again! – rationalbeing Jun 04 '16 at 13:51
@Servaes Not true, modular fractions are well defined as long as all the fractions can be represented with denominator coprime to the modulus, e.g. see this answer. – Bill Dubuque Jun 20 '16 at 21:39

Division in modular arithmetic

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