Numbers Between 1 and 100 Not Divisible by 3 or 5: A Comprehensive Analysis

Understanding how to count numbers within a specific range that are not divisible by certain integers can be a valuable mathematical skill. In this article, we delve into the specifics of determining how many numbers between 1 and 100 are not divisible by 3 and 5. This task involves the application of various mathematical techniques, including the usage of arithmetic sequences and the inclusion-exclusion principle.

Introduction

Divisibility is a fundamental concept in number theory. When dealing with numbers within a range, determining how many of them satisfy certain divisibility conditions can be both intriguing and practical, especially in applications such as networking, cryptography, and algorithm design.

Understanding the Problem

We are tasked with finding how many integers between 1 and 100 are not divisible by both 3 and 5. To solve this, we need to count the numbers divisible by 3, the numbers divisible by 5, and then apply the inclusion-exclusion principle to correct for the overlap (numbers divisible by both 3 and 5).

Step 1: Counting Multiples of 3

To find the multiples of 3 between 1 and 100, we start by identifying the first and last terms of this sequence:

The first term (a) is 3, and the last term (l) is 99. Using the arithmetic sequence formula l a (n - 1)d, we can solve for n:

99 3 (n - 1)3

Solving for n gives:

99 3 3n - 3
99 3n
33 n

Therefore, there are 33 multiples of 3 between 1 and 100.

Step 2: Counting Multiples of 5

Similarly, we count the multiples of 5 between 1 and 100. Here, the first term (a) is 5, and the last term (l) is 100:

Using the same formula, we find:

100 5 (n - 1)5

Solving for n gives:

100 5 5n - 5
100 5n
20 n

Hence, there are 20 multiples of 5 between 1 and 100.

Step 3: Counting Multiples of 15

Numbers that are divisible by both 3 and 5 are also divisible by 15. We now count the multiples of 15 between 1 and 100:

The first term (a) is 15, and the last term (l) is 90. Using the formula again, we find:

90 15 (n - 1)15

Solving for n gives:

90 15 15n - 15
90 15n
6 n

Therefore, there are 6 multiples of 15 between 1 and 100.

Applying the Inclusion-Exclusion Principle

To find the numbers that are divisible by either 3 or 5, we apply the inclusion-exclusion principle:

N(3 or 5) N(3) N(5) - N(15)

Substituting the values we found:

N(3 or 5) 33 20 - 6 47

So, there are 47 numbers between 1 and 100 that are divisible by 3 or 5.

Numbers Not Divisible by 3 or 5

Finally, to find the numbers that are not divisible by 3 or 5, we subtract the count of numbers divisible by 3 or 5 from the total count of numbers between 1 and 100:

N(not 3 or 5) 100 - 47 53

Therefore, there are 53 numbers between 1 and 100 that are not divisible by 3 or 5.

Conclusion

By applying the principles of arithmetic sequences and the inclusion-exclusion principle, we have determined that there are 53 numbers between 1 and 100 that are not divisible by 3 or 5. This analysis highlights the importance of these mathematical tools in solving complex counting problems.

Keywords: arithmetic sequences, inclusion-exclusion principle, multiples of 3 and 5