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Chapter 2: Problem 6

Let \(R(m, n)\) be the predicate "lf \(m\) is a factor of \(n^{2}\) then \(m\) is a factor of \(n, "\) with domain for both \(m\) and \(n\) being the set \(\mathbf{Z}\) of integers. a. Explain why \(R(m, n)\) is false if \(m=25\) and \(n=10\). b. Give values different from those in part (a) for which \(R(m, n)\) is false. c. Explain why \(R(m, n)\) is true if \(m=5\) and \(n=10\).

Short Answer

Expert verified
a. R(m, n) is false for m=25, n=10 because 25 is a factor of \(10^2 = 100\), but not a factor of 10. b. R(m, n) is false for m=9, n=6 because 9 is a factor of \(6^2 = 36\), but not a factor of 6. c. R(m, n) is true for m=5, n=10 because 5 is a factor of \(10^2 = 100\) and also a factor of 10.

Step by step solution

01

Part a: Explain why R(m, n) is false if m=25 and n=10

First, we need to check if 25 is a factor of 10^2. To do this, we will compute 10^2, and then check if it is divisible by 25. \(10^2 = 100\) Now, to check whether 25 is a factor of 100, we need to find the result of \(100 / 25\). Since \(100 / 25 = 4\), 25 is indeed a factor of 100. Now we need to check if 25 is also a factor of 10. If it were, then R(m, n) would be true. But we know that \(10 / 25 = 0.4\), which is not an integer. Thus, 25 is not a factor of 10. Since 25 is a factor of 10^2 but not a factor of 10, R(m, n) is false for m = 25 and n = 10.
02

Part b: Give values different from those in part (a) for which R(m, n) is false

Consider m=9 and n=6. We need to check if 9 is a factor of 6^2, and if 9 is a factor of 6. \(6^2 = 36\) To check if 9 is a factor of 36, we find the result of \(36 / 9\). Since \(36 / 9 = 4\), 9 is indeed a factor of 36. Now we need to check if 9 is also a factor of 6. Since \(6 / 9 = 2/3\), which is not an integer, 9 is not a factor of 6. Since 9 is a factor of 6^2 but not a factor of 6, R(m, n) is false for m = 9 and n = 6.
03

Part c: Explain why R(m, n) is true if m=5 and n=10

First, we need to check if 5 is a factor of 10^2. To do this, compute 10^2, and then check if it is divisible by 5. \(10^2 = 100\) Now, to check whether 5 is a factor of 100, we need to find the result of \(100 / 5\). Since \(100 / 5 = 20\), 5 is indeed a factor of 100. Now we need to check if 5 is also a factor of 10. If it is, R(m, n) will be true. Since \(10 / 5 = 2\), 5 is indeed a factor of 10. Since 5 is a factor of 10^2 and also a factor of 10, R(m, n) is true for m = 5 and n = 10.

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