Solving system of quadratic congruences

Question

If you have a system ex:

$ab \equiv 1 \mod 9$

$ab \equiv 3 \mod 10$

$ab \equiv 10 \mod 11$

$ab \equiv 7 \mod 12$

is there a way to determine integers $a$ and $b$?

Answer 1

The short answer is "No" - you can use the Chinese remainder theorem to find the equivalence class of $ab \pmod{d}$, where $d$ is the lcm of your moduli, but you cannot find the values of $a$ and $b$ individually.

For example, the congruence $ab \equiv 1 \pmod{9}$ has several solutions, as does $ab \equiv 3 \pmod{10}$. You can deduce that $ab \equiv 73 \pmod{90}$, but there will be many possible pairs of values for $a$ and $b$ which will make this true.