In the previous section, you found the conditions required for the pair of congruences
a1 (mod m1)
and
x
a2 (mod m2)
to have a unique solution x modulo m1m2. We now turn to the more general question of when this pair of congruences will have any solutions, unique or otherwise.
Research Question 2For what values of a1, a2, m1, and m2 will the pair of congruences
a1 (mod m1)
and
x
a2 (mod m2)
have a solution modulo m1m2? If there is a solution x0 (mod m1m2), find the form of all other solutions x (mod m1m2) in terms of x0. |
Below is our applet for your use in investigating Research Question 2.