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Unformatted text preview: x | y and x and y are positive, then x y . Theorem 2. If x and y are positive integers, then there are integers P and Q satisfying Px + Qy = gcd( x,y ) . Theorem 3. If n is an integer and a is a positive integer, then there is exactly one pair of integers ( q,r ) satisfying n = qa + r and r < a. Furthermore a | n if and only if r = 0 . Theorem 4. For each positive integer n let P ( n ) be a statement. If P (1) is true, and if P ( k + 1) is true whenever P ( k ) is true, then P ( n ) is true for all positive integers n ....
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- Fall '07