Two Equations:
ab % c = d
(ci + d) % c = d, i $\in \mathbb N$
I want to solve for b given the above two equations with a, c, and d known.
ab = ci + d
b = (ci + d) / a
i = (k + an), n $\in \mathbb Z$: n >= 0
b = (c(k+an) + d) / a
Again, since a, c, and d are known - and n can be any Integer >= 0 - the matter is solving for k.
Any advice?
An example.
(29 * 77) % 72 = 1
(72 * (31 + 77 * 0) + 1) = 2233 and 2233 % 72 = 1
(72 * (31 + 77 * 1) + 1) = 7777 and 7777 % 72 = 1
(72 * (31 + 77 * 2) + 1) = 13321 and 13321 % 72 = 1
$ci+d \equiv d \pmod c \implies ci \equiv 0 \pmod d$. Then choose $i$ such that $\gcd(i,d) = 1$ so it follows that $c = dk$ for some integer $k$.
Then $ab \equiv d \pmod c$ or $ab \equiv d \pmod dk$. Then $d|ab$ is the only information that we can gather without extra conditions.