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I am trying to understand if I could know something about the following relationship:

If I have:

$b \equiv n \mod a$

$d \equiv n \mod b$

$n \gt 0$

Is it possible to know something about the direct relationship of $a$ and $d$ or a combination of both?

$d \equiv ??? \mod a$

or other combinations, for instance, like:

$bd \equiv ???\mod ab$

I have been trying trial-error samples, and I am not sure if the usual rules of modular arithmetic could be applied somehow to get that direct relationship, so I am getting lost. Any hint or help is very appreciated, thank you!

UPDATE:

By reviewing directly the definitions of modularity:

$b = a*k_1+n,\ k_1 \in \Bbb N, k_1 \gt 0$

$d = b*k_2+n,\ k_2 \in \Bbb N, k_2 \gt 0$

thus,

$d = (a*k_1+n)*k_2+n = a*k_1*k_2+n*k_2+n = a*k_3+n*(k_2+1)$

$k_1*k_2=k_3 \in \Bbb N, k_3 \gt 0$

finally:

$d \equiv n(k_2+1) \mod a$

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Use the definitions:

$$b \equiv n \pmod a \iff \ \exists \ p \ \in \mathbb Z \ / \ b = pa + n$$

And:

$$d \equiv n \pmod b \iff \ \exists \ q \ \in \mathbb Z \ / \ d = qb + n$$

If you want to discover a property, proceed from here. Use techniques like comparing the different expressions of $n$, multiplying the two equations with each other, etc.

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  • $\begingroup$ yes you are right... I was just trying to apply to standard modular arithmetic rules, but I did not find anything through them... but studying directly the definitions I did found a relationship. I will update the question. $\endgroup$ – iadvd Apr 7 '15 at 2:25
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If $a$ and $b$ are relatively prime, then you can say something about bd mod ab.

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  • $\begingroup$ yes they are coprime, indeed $n \neq 0$ I will update the information. If they are coprime, please what it is possible to know? thank you! $\endgroup$ – iadvd Apr 7 '15 at 1:26
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    $\begingroup$ I think I missed something, sorry. In general, if a,b are coprime, and $ c \equiv d$ mod a and $c \equiv d $ mod b, then $c \equiv d$ mod ab. $\endgroup$ – mich95 Apr 7 '15 at 1:30
  • $\begingroup$ yes but that is not the same case that I have now because in the left side of my case $b \neq d$ $\endgroup$ – iadvd Apr 7 '15 at 1:37

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