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Chapter 8: Problem 6

An integer is called squarefree if it is not divisible by the square of a positive integer greater than \(1 .\) Find the number of squarefree positive integers less than 100 .

Short Answer

Expert verified
There are 71 square-free positive integers less than 100.

Step by step solution

01

- Identify squares of integers greater than 1

List the squares of positive integers greater than 1 that are less than 100. These squares are 4, 9, 16, 25, 36, 49, 64, and 81.
02

- Count multiples of each square

Count the multiples of each square number that are less than 100. The multiples are:- 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96 (25 multiples)- 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 (11 multiples)- 16: 16, 32, 48, 64, 80, 96 (6 multiples)- 25: 25, 50, 75 (3 multiples)- 36: 36, 72 (2 multiples)- 49: 49, 98 (2 multiples)- 64: 64 (1 multiple)- 81: 81 (1 multiple)
03

- Use the inclusion-exclusion principle

Since some numbers are counted multiple times, use the inclusion-exclusion principle to avoid double-counting.- Multiples of 4 and 9: 36, 72 (2 multiples)- Multiples of 4 and 16: 64, 96 (2 multiples)- Adjust for these overlaps.
04

- Calculate and adjust for overlaps

Adjust the counts by subtracting the overlaps to find the unique multiples of squares less than 100. Total overlaps: 4 (there are 132 - 4 = 28 unique multiples).
05

- Find the number of squarefree integers

Subtract the number of unique multiples from 99 (since we are considering numbers from 1 to 99):99 - 28 = 71

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square divisors
A squarefree integer is a number that isn鈥檛 divisible by any perfect square greater than 1. To understand this, let's break it down into simple terms. A perfect square is a number that results from squaring an integer, such as 4 (which is 2 squared) or 9 (which is 3 squared). The concept of 'divisors' means you are looking for numbers that can be divided without a remainder. In this case, we are interested in perfect squares as divisors. For example, 12 is not squarefree since it can be divided by 4. However, 10 is squarefree because it cannot be divided by any perfect squares other than 1.
inclusion-exclusion principle
The inclusion-exclusion principle helps us count the number of elements in the union of overlapping sets more accurately. Imagine you have two overlapping sets: Set A and Set B. If you simply add the sizes of Set A and Set B, you count the overlap twice. Therefore, you need to subtract the overlap once to correct the count. In our context, this principle is crucial because the multiples of squares such as 4 and 9 overlap (e.g., 36 and 72). By using the inclusion-exclusion principle, we adjust our count to avoid this double-counting. Subtracting these overlaps ensures we get the exact number of unique multiples of the squares.
counting multiples
Counting multiples is key to determining which numbers are not squarefree. To count multiples of a given perfect square under 100, list all the multiples incrementally. For instance, to find multiples of 4 under 100: 4, 8, 12, 16, and so on, up to 96. Similarly, do this for other perfect squares like 9, 16, 25, etc. By counting these, you identify the numbers that undermine the squarefree status. After counting the multiples of each square and then correctly adjusting for overlaps (using the inclusion-exclusion principle), you subtract this from the total numbers under 100 (i.e., 99) to find the squarefree integers.

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