Let $g: \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_d$, where $d=\gcd(m,n)$. If $g: (a,b) \mapsto ab \bmod d$. Then this is supposedly well defined. I want to know how to show this mathematically, by equating two tuples together, but also need to know what role does the gcd play in making this map well defined. Thank you for your answer.
Asked
Active
Viewed 57 times
1
-
Note $a \equiv a' \pmod m \implies a \equiv a' \pmod d $ – Steven Alexis Gregory Oct 19 '16 at 17:23
2 Answers
1
You need to show that $(a,b)\equiv(a',b')$ implies $ab\bmod d=a'b'\bmod d$. Here, $(a,b)\equiv(a',b')$ means that $a\equiv a'\pmod m$ and $b\equiv b'\pmod n$.
user26857
- 52,094
Hagen von Eitzen
- 374,180
1
Hint $\ $ If $\ d\mid m,n,\ $ e.g. if $\, d = \gcd(m,n),\ $ then
$\qquad\ \begin{align} A&\equiv a\pmod m\\ B&\equiv b\pmod n\end{align}\,\Rightarrow\,$ $\begin{align} A&\equiv a\pmod d\\ B&\equiv b\pmod d\end{align}\,\Rightarrow\ AB\equiv ab\pmod d$
The first arrow follows by transitivity of "divides" e.g. $\ d\mid m\mid A\!-\!a\,\Rightarrow\, d\mid A\!-\!a$
The second arrow follows by the Congruence Product Rule.
Bill Dubuque
- 272,048