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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

Find the number of positive integers \(n\) where \(1 \leq n \leq 1000\) and \(n\) is not a perfect square, cube, or fourth power.

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