Number theory question

[yossigi]

Hi all,
I have a small question regarding modular arithmetic:

I have a number
y = (x mod m1) mod m2

and I know that gcd(m1, m2) = 1

is there anything smart I can say on z = x (mod m2)?

If that's helpful, I can choose the relation between m1 and m2, that is I can choose whether m2 < m1 or m1 < m2
(but I gcd(m1, m2) must still be 1).

 

[daon]

(mod m1) mod m2 ? I've never seen this done as an algebra topic. Creating equivalence classes on equivalence classes loses a lot of information. I'm not even sure if its a well-defined operation.

If you're just looking for the remainder and m1 <= m2, then the answer doesn't change, i.e. y=z. Otherwise, we have m2 divides y-x(mod m1), so y - (m2)p = x (mod m1), so m1 divides y-(m2)p-x, i.e. x = y-(m2)p - (m1)q for some p,q. Then mod both sides by m2. This isn't a pretty answer, because we don't know what q or x is. With the knowledge that (m1,m2)=1, you might be able to eliminate one variable. I don't see how two is possible though.