Please explain your answer. WILL RATE LIFESAVE!!!!! Let a be an integer and let

Question

Please explain your answer. WILL RATE LIFESAVE!!!!!

Let a be an integer and let b be a natural number. Then there exist integers q and r such that a = bq + r and 0 r

Explanation / Answer

As suggested, we let R = {a-bx | x in Z and a-bx>=0}. R must be nonempty, because letting x equal any integer less than (a/b) produces a value of a-bx that satisfies the definition of the set R. Since R is nonempty, we apply the well-ordering principle, and state that R contains a smallest element, namely r. Then r = a-bq for some integer q. Since a-bx>=0, we have r>=0. Now, let us assume that r>=b. Then, r = b+k for some natural number k, k=0. Thus, a-(b+1)q>=0, and this value k is also an element of R. We already have k

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