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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 Fall '08
 HAULLGREN
 Rational number, Penn State University

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