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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

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 \(2,3, \ldots, k ; d_{0}=1 .\) )

Determine the number of positive integers \(n, 1 \leq n \leq 2000\), that are a) not divisible by 2,3 , or 5 b) not divisible by \(2,3,5\), or 7 c) not divisible by 2,3 , or 5 , but are divisible by 7

a) Let \(A=\\{1,2,3, \ldots, 7\\}\). A function \(f: A \rightarrow A\) is said to have a fixed point if for some \(x \in A, f(x)=x\). How many one-to-one functions \(f: A \rightarrow A\) have at least one fixed point? b) In how many ways can we devise a secret code by assigning to each letter of the alphabet a different letter to represent it?

In how many ways can one arrange the letters in CORRESPONDENTS so that (a) there is no pair of consecutive identical letters? (b) there are exactly two pairs of consecutive identical letters? (c) there are at least three pairs of consecutive identical letters?

a) In how many ways can the letters in ARRANGEMENT be arranged so that there are exactly two pairs of consecutive identical letters? at least two pairs of consecutive identical letters? b) Answer part (a), replacing two with three.
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