In the previous section, you found the conditions required for the pair of congruences

xa_{1}(modm_{1}) andxa_{2}(modm_{2})to have a unique solution

xmodulom_{1}m_{2}. We now turn to the more general question of when this pair of congruences will have any solutions, unique or otherwise.

## Research Question 2

For what values of

a_{1},a_{2},m_{1}, andm_{2}will the pair of congruencesxa_{1}(modm_{1}) andxa_{2}(modm_{2})have a solution modulo

m_{1}m_{2}? If there is a solutionx_{0}(modm_{1}m_{2}), find the form of all other solutionsx(modm_{1}m_{2}) in terms ofx_{0}.Below is our applet for your use in investigating Research Question 2.

Section 7.1 | Section 7.2 | Section 7.3 | Section 7.4

Copyright © 2001 by W. H. Freeman and Company