Finding Integers Between 1 and 300 Divisible by 3 and 5, but Not by 7

Divisibility rules are fundamental tools in number theory, providing insights into the properties of integers. This article delves into the concept of finding integers divisible by 3 and 5 but not by 7 within the range of 1 to 300. By applying the least common multiple (LCM) and exclusionary techniques, we can identify the precise count.

Introduction to Divisibility and LCM

Divisibility refers to the property of an integer being divisible by another integer without leaving a remainder. The least common multiple (LCM) of two or more integers is the smallest number that is evenly divisible by each one of them. This concept is crucial in solving problems related to divisibility and factorization.

Understanding the Steps to Solve

To find the count of integers between 1 and 300 that are divisible by 3 and 5 but not by 7, we follow a systematic approach. The process involves identifying the multiples of 15 (the LCM of 3 and 5), then removing the multiples of 105 (the LCM of 15 and 7).

Step 1: Counting Multiples of 15

First, we find the multiples of 15 within the range from 1 to 300.

Formula: [n frac{300}{15} 20]

Thus, there are 20 multiples of 15 in this range, which are 15, 30, 45, ..., 300.

Step 2: Subtracting Multiples of 105

The next step is to exclude the multiples of 15 that are also divisible by 7, i.e., the multiples of 105 (the LCM of 15 and 7).

Formula: [n frac{300}{105} approx 2.857 implies 2]

This indicates that there are 2 multiples of 105 in the range, which are 105 and 210.

Step 3: Calculating the Final Count

Finally, we subtract the multiples of 105 from the multiples of 15 to get the count of integers that satisfy the given conditions.

Final Count: [20 - 2 18]

Therefore, there are 18 integers between 1 and 300 that are divisible by 3 and 5 but not by 7.

Conclusion

Using the principles of divisibility and the least common multiple, we can efficiently solve complex problems involving multiple factors. The systematic approach of identifying multiples and subsequent exclusions ensures accuracy and clarity in the solution. This method can be applied to various scenarios where counting specific integer sets is required.

Key Takeaways

Understanding divisibility rules is essential for solving number theory problems. The least common multiple (LCM) is a powerful tool for identifying common multiples of multiple integers. Systematic exclusion of unwanted multiples is crucial to finding the precise count of integers.

References

For further reading and deeper understanding, explore resources on number theory and divisibility in mathematics. This topic is foundational in computer science, cryptography, and various mathematical applications.