Counting the distinct values of a modular mapping

[ing]

Hello,
first of all, sorry if my question is either trivial or imprecise, I'm from the engineering domain :)

I need to know how many different values the following pair can take:

\left(a\cdot i + b\cdot j\right) \bmod n_1
\left(c\cdot i + d\cdot j\right) \bmod n_2

as (i,j) spans \mathbb{Z}^2, with given integers a, b, c, d, n_1, n_2.

I know that, in case I had a single expression, i.e.

\left(a\cdot i + b\cdot j\right) \bmod n

the answer would be \frac{n}{\gcd(\gcd(a,b),n)}.

I suspect that the answer to my question looks similar.
In particular, I was trying to establish an isomorphism between \left(\mathbb{Z}_{n_1},\mathbb{Z}_{n_2}\right) and \mathbb{Z}_{n_1\cdot n_2}, obtaining something looking like

\left(u\cdot i + v\cdot j\right) \bmod \left( n_1\cdot n_2\right)

so as to exploit the same result, but so far I didn't come out with anything useful.

Any clues?
Thanks!

 

[Erland]

In particular, I was trying to establish an isomorphism between \left(\mathbb{Z}_{n_1},\mathbb{Z}_{n_2}\right) and \mathbb{Z}_{n_1\cdot n_2}

But that's not true in general. For example \left(\mathbb{Z}_2,\mathbb{Z}_2\right) is not isomorphic to \mathbb{Z}_4. In the former, all nonzero elements have order 2, while the latter has a generator of order 2.

(Unless I misunderstand your notation.)