how many integers from 1 to 1000 are mu

2 min read 31-01-2025

How Many Integers from 1 to 1000 are Multiples of 3, 5, or 7?

Finding the number of integers from 1 to 1000 that are multiples of 3, 5, or 7 requires a systematic approach using the principles of inclusion-exclusion. This seemingly simple problem introduces us to important concepts in number theory and combinatorics. Let's break down the solution step-by-step.

Understanding the Problem

Our goal is to determine the count of integers between 1 and 1000 (inclusive) that are divisible by at least one of the numbers 3, 5, or 7. We can't simply add the number of multiples of each individually, as this would lead to overcounting. For example, a number divisible by both 3 and 5 (like 15) would be counted twice. The inclusion-exclusion principle helps us correct for this overcounting.

Applying the Inclusion-Exclusion Principle

The inclusion-exclusion principle states that for sets A, B, and C:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Where:

|A ∪ B ∪ C| represents the total number of elements in the union of sets A, B, and C.
|A|, |B|, |C| represent the number of elements in each individual set.
|A ∩ B|, |A ∩ C|, |B ∩ C| represent the number of elements in the intersection of two sets.
|A ∩ B ∩ C| represents the number of elements in the intersection of all three sets.

In our case:

A = Set of multiples of 3 between 1 and 1000
B = Set of multiples of 5 between 1 and 1000
C = Set of multiples of 7 between 1 and 1000

Calculating the Individual Set Sizes

First, let's find the number of elements in each set:

|A| (Multiples of 3): 1000 divided by 3 is approximately 333.33. Since we only want whole numbers, there are 333 multiples of 3.
|B| (Multiples of 5): 1000 divided by 5 is 200. There are 200 multiples of 5.
|C| (Multiples of 7): 1000 divided by 7 is approximately 142.86. There are 142 multiples of 7.

Calculating the Intersection Sizes

Next, we need to find the number of elements in the intersections:

|A ∩ B| (Multiples of both 3 and 5, i.e., multiples of 15): 1000 divided by 15 is 66.67, so there are 66 multiples of 15.
|A ∩ C| (Multiples of both 3 and 7, i.e., multiples of 21): 1000 divided by 21 is approximately 47.62, so there are 47 multiples of 21.
|B ∩ C| (Multiples of both 5 and 7, i.e., multiples of 35): 1000 divided by 35 is approximately 28.57, so there are 28 multiples of 35.
|A ∩ B ∩ C| (Multiples of 3, 5, and 7, i.e., multiples of 105): 1000 divided by 105 is approximately 9.52, so there are 9 multiples of 105.

Applying the Inclusion-Exclusion Formula

Now, we can plug these values into the inclusion-exclusion formula:

|A ∪ B ∪ C| = 333 + 200 + 142 - 66 - 47 - 28 + 9 = 543

Conclusion

Therefore, there are 543 integers from 1 to 1000 that are multiples of 3, 5, or 7. This problem highlights the importance of the inclusion-exclusion principle in accurately counting elements in overlapping sets. This technique is valuable in various fields, including probability and combinatorics.

how many integers from 1 to 1000 are mu