Counting Natural Numbers Divisible by 3 or 5 Between 1 and 400

Understanding how to count natural numbers divisible by either 3 or 5 between a specific range is a fundamental concept in number theory. This article aims to break down the process step-by-step, providing a clear understanding of the methods and formulas used. By the end, you will be equipped with the skills to solve similar problems efficiently.

Introduction

This article focuses on identifying and counting the numbers between 1 and 400 that can be divisible by both 3 and 5. We will first delve into the specifics and then extend the solution to finding numbers divisible by either 3 or 5 within the same range.

Numbers Divisible by 3 and 5

First, let's consider the numbers that are divisible by both 3 and 5 within the range of 1 to 400. The least common multiple (LCM) of 3 and 5 is 15. Therefore, the numbers that can be divisible by both 3 and 5 are the multiples of 15 within the specified range.

To find how many such numbers exist between 1 and 400, we can use the formula for the number of terms in an arithmetic progression (AP):

[T_n a (n-1)d]

For the multiples of 15, the first term (a 15), the common difference (d 15), and the last term (T_n 390). Plugging these values into the formula, we can solve for (n):

[390 15n - 15n - 15 150]

Thus, we have:

[390 15n - 15]

Let's solve for (n):

[15n 405]

[n frac{405}{15} 27]

So, the numbers divisible by both 3 and 5 between 1 and 400 are: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375, and 390.

Numbers Divisible by 3 or 5

Next, we will count the numbers that are divisible by either 3 or 5 within the range of 1 to 400. We can break this problem into three parts:

Count the numbers divisible by 3. Count the numbers divisible by 5. Subtract the numbers counted in both 3 and 5 to avoid double-counting.

Starting with numbers divisible by 3:

The first term is (a 3), the common difference is (d 3), and the last term is (T_n 399). Using the formula:

[399 3n - 13]

Solving for (n):

[399 3n - 3] [n frac{402}{3} 134]

This means there are 134 numbers divisible by 3 between 1 and 400.

Next, for numbers divisible by 5:

The first term is (a 5), the common difference is (d 5), and the last term is (T_n 400). Using the formula: [400 5n - 15] [n frac{405}{5} 81]

This means there are 81 numbers divisible by 5 between 1 and 400.

Finally, for the numbers divisible by both 3 and 5, as calculated earlier, there are 27 numbers.

The total count of numbers divisible by either 3 or 5 is then given by:

[134 81 - 27 188]

Conclusion

The process of counting numbers divisible by either 3 or 5 between 1 and 400 involves a clear and systematic approach using arithmetic progressions. By understanding the basics of LCM and the formula for the number of terms in an AP, you can solve similar problems efficiently. This method can be applied to a wide range of numerical problems, making it a valuable skill in mathematics and data analysis.