Numbers Divisible by 3 or 5: A Comprehensive Analysis
In this article, we will delve into the problem of determining how many numbers between 1 and 500 are divisible by either 3 or 5. This involves understanding the principles of divisibility, multiples, and prime factorization. We will break down the problem step-by-step, applying arithmetic progression formulas and considering the Least Common Multiple (LCM) to find the accurate count.
Understanding the Basics of Divisibility
A number is divisible by another if it can be divided by it without leaving a remainder. When a number is divisible by both 3 and 5, it is known to be a multiple of their Least Common Multiple (LCM), which in this case is 15.
Divisibility by 15
If a number is divisible by both 3 and 5, it is also divisible by 15. This is because 15 is the LCM of 3 and 5, and any number that contains both 3 and 5 as factors will inherently be divisible by 15.
Counting Numbers Divisible by 3 or 5 from 1 to 500
Let's start by determining how many numbers between 1 and 500 are divisible by 3 or 5.
Direct Count for Numbers Divisible by 3
The sequence of numbers divisible by 3 from 1 to 498 can be described as an arithmetic sequence where the first term (a) is 3, the common difference (d) is 3, and the last term (T_n) is 498. Using the formula for the (n)-th term of an arithmetic progression (AP), we have:
[T_n a (n-1)d]
[498 3 (n-1) cdot 3]
[498 3n]
[n frac{498}{3} 166]
So, there are 166 numbers between 1 and 500 that are divisible by 3.
Direct Count for Numbers Divisible by 5
The sequence of numbers divisible by 5 from 1 to 500 can be described as an arithmetic sequence where the first term (a) is 5, the common difference (d) is 5, and the last term (T_n) is 500. Using the formula for the (n)-th term of an AP, we have:
[T_n a (n-1)d]
[500 5 (n-1) cdot 5]
[500 5n]
[n frac{500}{5} 100]
So, there are 100 numbers between 1 and 500 that are divisible by 5.
Adjusting for Overlap (Numbers Divisible by 15)
When we count numbers divisible by 3 and 5 separately, we double-count the numbers that are divisible by both. These numbers are the multiples of 15. Let's find how many multiples of 15 are there between 1 and 500:
The sequence of numbers divisible by 15 from 1 to 495 can be described as an arithmetic sequence where the first term (a) is 15, the common difference (d) is 15, and the last term (T_n) is 495. Using the formula for the (n)-th term of an AP, we have:
[T_n a (n-1)d]
[495 15 (n-1) cdot 15]
[495 15n]
[n frac{495}{15} 33]
So, there are 33 numbers between 1 and 500 that are divisible by 15.
Final Calculation
To find the total number of numbers between 1 and 500 that are divisible by 3 or 5, we use the principle of inclusion-exclusion:
Total (Numbers divisible by 3) (Numbers divisible by 5) - (Numbers divisible by 15)
Total 166 100 - 33 233
Therefore, there are 233 numbers between 1 and 500 that are divisible by either 3 or 5.
Examples to Illustrate the Concept
Let's consider a smaller range, for instance, numbers from 1 to 20, to illustrate the concept of exclusion of multiples of 15.
Example 1: Numbers from 1 to 20
Numbers divisible by 3: 3, 6, 9, 12, 15, 18 (6 numbers)
Numbers divisible by 5: 5, 10, 15, 20 (4 numbers)
Number 15 is counted twice, so the correct count is 6 4 - 1 9 numbers.
Example 2: Numbers from 1 to 400
Numbers divisible by 3: 3, 6, 9, ..., 399 (133 numbers)
Numbers divisible by 5: 5, 10, 15, ..., 400 (80 numbers)
Numbers divisible by 15: 15, 30, 45, ..., 390 (26 numbers)
Thus, the total number of numbers divisible by 3 or 5 is 133 80 - 26 187.
This example confirms our formula and approach.
Conclusion
In this comprehensive analysis, we have used fundamental properties of arithmetic progressions, inclusion-exclusion principle, and the concept of multiples to find the accurate count of numbers divisible by 3 or 5 within a given range. This methodology can be applied to any range and can help in understanding the underlying principles of divisibility and multiples.