In the congruences in Exercises , and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo .
Whether there are solutions of congruence . If there are, find incongruent solutions modulo .
i) For , where if there is a solution to , then divides .
ii) Suppose that and that is divisor of . Let and , where are integers the following statements are hold:
a) , if and only if .
b) If and are any two solutions of , then it is follows that .
c) Let be a fixed solution to , and each of the integers in the list
is a solution to .
d) No two solutions listed in part c) are congruent modulo .
e) Any solution to congruent to one of the numbers listed in part c).
iii) The Euclidean Algorithm:
Since the integers are decreasing and are all non-negative, there is a smallest integer , such that : .
iv) Theorem: Addition and Multiplication Properties:
If and is any integer, then
By adding property of congruence’s,
Comparing with .
By using result, for where if there is a solution to , then divides .
Thus, there are exactly incongruent solutions modulo
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