Modular arithmetic calculation

Question

Is there any way to calculate (a mod c)*(b mod c)?

Knowing only:

• a*b
• c
• d
• ((a mod c)*(b mod c)) mod c
• (a mod d)*(b mod d)

Answer 1

No, it's impossible.

Here's a counterexample:

ab = 1225 = (5)(5)(7)(7)
c = 3
d = 5000
((a mod c)(b mod c)) mod c = 1
(a mod d)(b mod d) = 1225

If a=25 and b=49, then (a mod c)(b mod c)=(1)(1)=1

If a=35 and b=35, then (a mod c)(b mod c)=(2)(2)=4