a) Show that the system of congruences.x: ? a2(modm1) and x ? a2 (mod m2), where a1, a2. m1, and m2 are integers with m1 >0 and m2 > 0, has a solution if and only if gcd(m1, m2) | a1 – a2.________________b) Show that if the system in part (a) has a solution, then it is unique modulo 1cm(m1, m2).
Solution Step 1:In part (a) we have to show that system of congruences and Where are integers with and ,has a solution if and only if Step 2:(a)The system of congruences has solution iff such that Rearranging we get Since is linear combination of and .any integer which is linear combination of and will be multiple of gcd(,) and hence gcd(,must divides .Now consider gcd(, divides that is let By extension of Euclid’s algorithm there exists p and q such that Hence and is solution for and hence is solution to the system.