Division with Remainder, Infinitude of Primes

Division with Remainder, Infinitude of Primes - Math 100...

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Math 100 Three Proofs by Induction Martin H. Weissman 1. Division with Remainder Theorem 1. Suppose that x and y are natural numbers, and y 6 = 0 . Then there exist natural numbers q and r such that: x = qy + r . 0 r < y . Proof. We prove the theorem, by induction on x . When x = 0, let q = 0, and r = 0. Then, we find that x = qy + r , and 0 r < y as desired. Hence the result is proven when x = 0. Now, assume that x > 0, and assume that the theorem has been proven for all smaller values of x . Then, there are two cases to consider: either x < y or x y . If x < y , then let q = 0 and r = x . Then, we find that x = qy + r and 0 r < y as desired. Hence the result is proven when x < y . If x y , then let x 0 = x - y . Observe that 0 x 0 < x . Hence, there exists q 0 ,r N , such that x 0 = q 0 y + r and 0 r < y . Since x = x 0 + y , we find that x = q 0 y + y + r . Let q = q 0 + 1. Then, x = qy + r. The result is proven when x y . By induction, the theorem is proven for all x and y . / For the next result, it is useful to have the following basic principle: Theorem 2. Suppose that x Z , q N , and x is a multiple of q . Then x = 0 or | x | ≥ q . Proof.
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Division with Remainder, Infinitude of Primes - Math 100...

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