91Ó°ÊÓ

Use the properties of even and odd integers that are listed in Example \(3.2 .3\) to do Indicate which properties you use to justify your reasoning.True or false? If \(a\) is any odd integer, then \(a^{2}+a\) is even. Explain.

Prove that a necessary and sufficient condition for a nonnegative integer \(n\) to be divisible by a positive integer \(d\) is that \(n\) mod \(d=0\).

For all real numbers \(x\), if \(x>1\) then \(x^{2}>x\).

For all integers \(m\) and \(n\), if \(m n=1\) then \(m=n=1\) or \(m=n=-1\).

Use the proof technique illustrated in exercise \(21(b)\) to prove that if \(n\) is any integer that is not a perfect square, then \(\sqrt{n}\) is irrational.

A matrix \(\mathbf{M}\) has 3 rows and 4 columns.$$ \left[\begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{array}\right] $$The 12 entries in the matrix are to be stored in row major form in locations 7,609 to 7,620 in a computer's memory. This means that the entries in the first row (reading left to right) are stored first, then the entries in the second row, and finally the entries in the third row. a. Which location will \(a_{22}\) be stored in? b. Write a formula (in \(i\) and \(j\) ) that gives the integer \(n\) so that \(a_{i j}\) is stored in location \(7,609+n\). c. Find formulas (in \(n\) ) for \(r\) and \(s\) so that \(a_{r s}\) is stored in location \(7,609+n\).

Prove each of the statements in 23-29. For any real number \(x\), if \(x\) is not an integer, then \(\lfloor x\rfloor+\lfloor-x\rfloor=-1 .\)

Prove that the product of any two consecutive integers is even.

An alternative to the Euclidean algorithm uses subtraction rather than division to compute greatest common divisors. (After all, division is repeated subtraction.) It is based on the following lemma: Lemma 3.8.3 If \(a \geq b>0\), then \(\operatorname{gcd}(a, b)=\operatorname{gcd}(b, a-b)\) a. Prove Lemma \(3.8 .3 .\) b. Trace the execution of Algorithm \(3.8 .3\) for \(A=630\) and \(B=336\). c. Trace the execution of Algorithm \(3.8 .3\) for \(A=768\) and \(B=348\).

Prove that \(\log _{5}(2)\) is irrational.