Finish the proof of Theorem 8.4.5 by proving that if a,b, and c are as in the proof, then .
are integers then .
are integers such that for some integers and .
Here, is the least element in .
Let be an integer and be a positive integer. Then, there are unique integers with such that , here, is the quotient and is the remainder.
So, and .
divides if there exists an integer such that . The notation is
By the division algorithm, there exist integers with such that .
Subtract from both side of equation
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