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Magazine Problem 1
Assume that \(k\) is a particular integer. a. Is \(-17\) an odd integer? b. Is 0 an even integer? c. Is \(2 k-1\) odd?
Problem 1
A calculator display shows that \(\sqrt{2}=1.414213562\), and \(1.414213562=\frac{1414213562}{1000000000}\). This suggests that \(\sqrt{2}\) is a rational number, which contradicts Theorem 3.7.1. Explain the discrepancy.
Problem 1
Give a reason for your answer in each of 1-13. Assume that all variables represent integers. Is 52 divisible by 13 ?
Problem 1
For each of the values of \(n\) and \(d\) given in \(1-6\), find integers \(q\) and
\(r\) such that \(n=d q+r\) and \(0 \leq r
Problem 2
Is \(\frac{1}{0}\) an irrational number? Explain.
Problem 2
Write each number as a ratio of two integers.\(4.6037\)
Problem 2
Assume that \(m\) and \(n\) are particular integers. \(\begin{array}{ll}\text { a. Is } 6 m+8 n \text { even? } & \text { b. Is } 10 m n+7 \text { odd? }\end{array}\) c. If \(m>n>0\), is \(m^{2}-n^{2}\) composite?
Problem 2
Example 3.2.1(h) illustrates a technique for showing that any repeating decimal number is rational. A calculator display shows the result of a certain calculation as \(40.72727272727\). Can you be sure that the result of the calculation is a rational number? Explain.
Problem 3
Assume that \(r\) and \(s\) are particular integers. a. Is \(4 r s\) even? b. Is \(6 r+4 s^{2}+3\) odd? c. If \(r\) and \(s\) are both positive, is \(r^{2}+2 r s+s^{2}\) composite?
Problem 3
Use proof by contradiction to show that for all integers \(n\), \(3 n+2\) is not divisible by 3 .
