For any integer and positive integer , there exist unique integers and such that and .
Existence: Let The intersection of the sets and is non-empty and has a positive element (take ). By the Well Ordering Principle , it contains a least element. Let equal the value of the least element and let equal the respective value of . Therefore, . Now we prove that by contradiction. Let . Then . However, this leads to a contradiction ( is supposed to be the smallest positive value that can be expressed in the form , but is smaller, positive, and can also be expressed in this manner). Therefore, .
Uniqueness: Let . Then . Then is a multiple of . However, since and , then . The only multiple of that can equal is . Therefore, . Since , .
If and , then we say or " divides ". Note that every integer divides .
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