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Chapter 8: Problem 1

Determine how many \(n \in \mathbf{Z}^{+}\)satisfy \(n \leq 500\) and are not divisible by \(2,3,5,6,8\), or 10 .

Short Answer

Expert verified
The number of \(n \in \mathbf{Z}^{+}\)that satisfy \(n \leq 500\) and are not divisible by \(2,3,5,6,8\), or 10 is equivalent to the rounded result of the calculation from Step 2.

Step by step solution

01

Understand the nature of numbers

Initial step involves recognizing the pattern in the provided numbers. The numbers to check for divisibility are 2, 3, 5, 6, 8, and 10. However, it's important to note that some of these numbers share common factors (multiple of each other). The only unique prime factors involved are 2, 3, and 5.
02

Calculate totatives

We can exclude larger numbers divisible by 2, 3, and 5 from our possible options with a simple trick. To calculate the number of positive integers less than 500, which are not divisible by 2, 3, or 5, we use the multiplicative function of totatives (also called Euler's totient function). As per function rule, \( \phi(500) \) = \( 500 \times (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{5}) \). Calculate this to find number of totatives.
03

Round and Interpret

The result of Step 2 will most likely be a decimal. Round it to the nearest whole number to find the answer. Be sure to understand that this number reflects the positive integers less than or equal to 500 that are not divisible by 2, 3, or 5.

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Most popular questions from this chapter

Compute \(\phi(n)\) for \(n\) equal to (a) 5186 ; (b) 5187 ; (c) 5188 .

Ms. Pezzulo teaches geometry and then biology to a class of 12 advanced students in a classroom that has only 12 desks. In how many ways can she assign the students to these desks so that (a) no student is seated at the same desk for both classes? (b) there are exactly six students each of whom occupies the same desk for both classes?

Give a combinatorial argument to verify that for all \(n \in \mathbf{Z}^{+}\), $$ n !=\left(\begin{array}{l} n \\ 0 \end{array}\right) d_{0}+\left(\begin{array}{l} n \\ 1 \end{array}\right) d_{1}+\left(\begin{array}{l} n \\ 2 \end{array}\right) d_{2}+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right) d_{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) d_{k} $$ (For each \(1 \leq k \leq n, d_{k}=\) the number of derangements of 1 \(2,3, \ldots, k ; d_{0}=1 .\) )

In how many ways can one arrange all of the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? [Here we want to count arrangements such as IINNOOFRMTA and FORTMAIINON but not INFORINMOTA (where "IN" occurs twice) or NORTFNOIAMI (where "NO" occurs twice).]

a) List all the derangements of \(1,2,3,4,5\) where the first three numbers are 1,2, and 3 , in some order. b) List all the derangements of \(1,2,3,4,5,6\) where the first three numbers are 1,2 , and 3, in some order.
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