A Number Theory Reduction

Question

If we know the value of $a\pmod {p_1}$ and $b\pmod {p_2}$ where $p_1$ and $p_2$ are prime, is it possible to find the value of $ab\pmod {p_1p_2}$ ? . I was thinking of using the chinese remainder theorem but I can't seem to generalize it.

Answer 1

Since you mention CRT, I assume $p_{1} \ne p_{2}$.

The answer is no.

Think of $a = 1 + s p_{1}$ (all values are congruent modulo $p_{1}$), $b = 1 + t p_{2}$ (all values are congruent modulo $p_{2}$), which yield all the products $$ ab \equiv 1 + s p_{1} + t p_{2} \pmod{p_{1} p_{2}}. $$

For instance when $p_{1} = 3$ and $p_{2} = 5$ we have $$ 1 \cdot 1 = 1 \not\equiv 9 \equiv (1 + 3) (1 + 5) \pmod{15}. $$