It's been a while, but I'll take a stab at it anyway.

"r1" and "r2" are classes of numbers.

ex. r1 = { ... r1 - 2x, r1 - x, r1, r1 + x, r1 +2x ...}

so,

N mod x = r1 <=> r1 + nx = N, for n integer (1) (def'n)

N mod y = r2 <=> r2 + my = N, for m integer (2) (def'n)

(with n not necessarily equal to m)

x*y = [(N-r1)(N-r2)]/nm, by (1) and (2)

Let p = N mod(x*y) <=> p + l [[N-r1)(N-r2)]/nm] = N (def'n), for l integer.

then p = N - (l/nm)[(N-r1)(N-r2)]