hw3 - (2 q 1 = 2 r where r is an integer By definition of...

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CMPSC 360 Discrete Mathematics for Computer Science Fall 2009 Penn State University Assignment #3 Due: 14 September 1. Prove that there are distinct integers m and n such that 2 m - 3 n is an integer. 2. Prove that there is an integer k such that 4 k 2 - 17 x - 7 is a power of 2. 3. Prove that if any numbers a and b are odd then a + b + 2 is even. 4. Prove that the difference between an even number and an odd number is odd. 5. Find the mistake in the following “proof” of the statement “The product of an even integer and an odd integer is even.” Suppose m is an even integer and n is an odd integer. If m · n is even, then by definition of even there exists an integer r such that m · nn = 2 r . Also since m is even, there exists an integer p such that m = 2 p and since n is odd there exists an integer q such that n = 2 q + 1. Thus m · n = (2 p
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Unformatted text preview: ) + (2 q + 1) = 2 r where r is an integer. By definition of even, then, m · n is even. 6. Prove that for all integers p,q > 0, if p and q are perfect squares then so is pq . (An integer is a perfect square iff it is equal to the square of some integer.) 7. Suppose n is odd. Is it the case that n 2 + n must be even? If it is, then give a proof. If not, then give a counterexample. 8. Prove that any rational number raised to an integer power is a rational number. 9. Prove that if a and b are any two rational numbers with a < b then there exists another rational number c such that a < c < b . Is your proof constructive? Explain. 10. Is the reciprocal of any rational number a rational number? Explain....
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