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!
By reviewing directly the definitions of modularity:
$b = a*k_1+n,\ k_1 \in \Bbb N, k_1 \gt 0$
$d = b*k_2+n,\ k_2 \in \Bbb N, k_2 \gt 0$
$d = (a*k_1+n)*k_2+n = a*k_1*k_2+n*k_2+n = a*k_3+n*(k_2+1)$
$k_1*k_2=k_3 \in \Bbb N, k_3 \gt 0$
$d \equiv n(k_2+1) \mod a$