Tests - x | y and x and y are positive, then x y . Theorem...

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MATH 74 FINAL EXAM THEOREM SHEET You may take for granted that + and · are binary operations on Z satisfying the commutative, associative, and distributive laws. You thus do not need to pay attention to the placement of parentheses where these laws and the “order of operations” convention make an expression unambiguous. You may also assume the usual properties of the notion of and < on Z . You may assume I am familiar with the definition of “divides” in terms of multiplication. Theorem 1. If x,y,z are integers then (1) If x | y and y | z , then x | z . (2) If x | y , then xz | yz . (3) If x | y , then x | yz . (4) If x | y and x | z , then x | ( y + z ) and x | ( y - z ) . (5) If
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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 &lt; 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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