91Ó°ÊÓ

Prove that for all real numbers \(c\), if \(c\) is a root of a polynomial with rational coefficients, then \(c\) is a root of a polynomial with integer coefficients.

Use the results of exercises 28 and 30 to determine whether the following numbers are prime. a. 9,269 b. 9,103 c. 8,623 d. 7,917

There exists an integer \(n\) such that \(6 n^{2}+27\) is prime.

Two athletes run a circular track at a steady pace so that the first completes one round in 8 minutes and the second in 10 minutes. If they both start from the same spot at \(4 \mathrm{PM}\)., when will be the first time they return to the start together?

Prove that there exists a unique prime number of the form \(n^{2}-1\), where \(n\) is an integer that is greater than or equal to 2 .

There exists an integer \(k\) such that \(k \geq 4\) and \(2 k^{2}-5 k+2\) is prime.

Given any integer \(n\), if \(n>3\), could \(n, n+2\), and \(n+4\) all be prime? Prove or give a counterexample.

"Proof: Suppose \(r\) and \(s\) are rational numbers. By definition of rational, \(r=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\), and \(s=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\). Then \(r+s=a / b+a / b=2 a / b\). Let \(p=2 a\). Then \(p\) is an integer since it is a product of integers. Hence \(r+s=p / b\), where \(p\) and \(b\) are integers and \(b \neq 0\). Thus \(r+s\) is a rational number by definition of rational. This is what was to be shown."

Theorem: The difference between any odd integer and any even integer is odd. "Proof: Suppose \(n\) is any odd integer, and \(m\) is any even integer. By definition of odd, \(n=2 k+1\) where \(k\) is an integer, and by definition of even, \(m=2 k\) where \(k\) is an integer. Then \(n-m=(2 k+1)-2 k=1 .\) But 1 is odd. Therefore, the difference between any odd integer and any even integer is odd."

"Proof: Suppose \(r\) and \(s\) are rational numbers. Then \(r=a / b\) and \(s=c / d\) for some integers \(a, b, c\), and \(d\) with \(b \neq 0\) and \(d \neq 0\) (by definition of rational). Then \(r+s=\) \(a / b+c / d\). But this is a sum of two fractions, which is a fraction. So \(r+s\) is a rational number since a rational number is a fraction."