How Many Integers From 1 To 1000 Are Mu

How Many Integers from 1 to 1000 are Multiples of 2, 3, 5, or 7?

Determining the number of integers from 1 to 1000 that are multiples of 2, 3, 5, or 7 requires a systematic approach leveraging the principles of set theory and number theory. We can't simply add the number of multiples of each individual number, as this would lead to significant overcounting due to overlapping multiples (e.g., a number could be a multiple of both 2 and 3). This problem is best solved using the Principle of Inclusion-Exclusion.

Understanding the Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion is a counting technique used to determine the cardinality (size) of a union of multiple sets. When dealing with sets that have overlapping elements, simply adding the sizes of each set will result in an inflated count. The principle corrects for this overcounting.

For two sets A and B, the formula is:

|A ∪ B| = |A| + |B| - |A ∩ B|

Where:

• |A ∪ B| represents the size of the union of sets A and B (elements in either A or B or both).
• |A| represents the size of set A.
• |B| represents the size of set B.
• |A ∩ B| represents the size of the intersection of sets A and B (elements in both A and B).

This principle extends to more than two sets. For three sets A, B, and C:

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

And so on for even more sets. The pattern alternates between addition and subtraction.

Applying the Principle to Our Problem

Let's define our sets:

• A: Set of integers from 1 to 1000 that are multiples of 2.
• B: Set of integers from 1 to 1000 that are multiples of 3.
• C: Set of integers from 1 to 1000 that are multiples of 5.
• D: Set of integers from 1 to 1000 that are multiples of 7.

Our goal is to find |A ∪ B ∪ C ∪ D|.

First, let's find the size of each individual set:

• |A|: The number of multiples of 2 from 1 to 1000 is ⌊1000/2⌋ = 500
• |B|: The number of multiples of 3 from 1 to 1000 is ⌊1000/3⌋ = 333
• |C|: The number of multiples of 5 from 1 to 1000 is ⌊1000/5⌋ = 200
• |D|: The number of multiples of 7 from 1 to 1000 is ⌊1000/7⌋ = 142

Now, we need to find the sizes of the intersections:

• |A ∩ B|: Multiples of both 2 and 3 (multiples of 6): ⌊1000/6⌋ = 166
• |A ∩ C|: Multiples of both 2 and 5 (multiples of 10): ⌊1000/10⌋ = 100
• |A ∩ D|: Multiples of both 2 and 7 (multiples of 14): ⌊1000/14⌋ = 71
• |B ∩ C|: Multiples of both 3 and 5 (multiples of 15): ⌊1000/15⌋ = 66
• |B ∩ D|: Multiples of both 3 and 7 (multiples of 21): ⌊1000/21⌋ = 47
• |C ∩ D|: Multiples of both 5 and 7 (multiples of 35): ⌊1000/35⌋ = 28

And finally, the intersection of all four:

• |A ∩ B ∩ C ∩ D|: Multiples of 2, 3, 5, and 7 (multiples of 210): ⌊1000/210⌋ = 4

Three-way intersections:

• |A ∩ B ∩ C|: Multiples of 6, 10, and 15 (multiples of 30): ⌊1000/30⌋ = 33
• |A ∩ B ∩ D|: Multiples of 6, 14, and 21 (multiples of 42): ⌊1000/42⌋ = 23
• |A ∩ C ∩ D|: Multiples of 10, 14, and 35 (multiples of 70): ⌊1000/70⌋ = 14
• |B ∩ C ∩ D|: Multiples of 15, 21, and 35 (multiples of 105): ⌊1000/105⌋ = 9

Now, applying the Principle of Inclusion-Exclusion:

|A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D| - (|A ∩ B| + |A ∩ C| + |A ∩ D| + |B ∩ C| + |B ∩ D| + |C ∩ D|) + (|A ∩ B ∩ C| + |A ∩ B ∩ D| + |A ∩ C ∩ D| + |B ∩ C ∩ D|) - |A ∩ B ∩ C ∩ D|

|A ∪ B ∪ C ∪ D| = 500 + 333 + 200 + 142 - (166 + 100 + 71 + 66 + 47 + 28) + (33 + 23 + 14 + 9) - 4

|A ∪ B ∪ C ∪ D| = 1175 - 478 + 79 - 4 = 772

Therefore, there are 772 integers from 1 to 1000 that are multiples of 2, 3, 5, or 7.

Alternative Approach using Euler's Totient Function (for advanced readers)

While the Principle of Inclusion-Exclusion is the most straightforward method for this problem, a more advanced approach involves using Euler's totient function, particularly if dealing with a larger range or more prime factors. This function helps to determine the number of integers less than or equal to n that are relatively prime to n. However, it's significantly more complex for this specific problem and less intuitive than the inclusion-exclusion method.

Conclusion

Determining the number of integers from 1 to 1000 that are multiples of 2, 3, 5, or 7 efficiently requires using the Principle of Inclusion-Exclusion. This powerful counting technique systematically accounts for overlapping multiples, preventing overcounting and providing an accurate result. Understanding this principle is crucial for tackling similar problems in combinatorics and number theory. The final answer, 772, represents the total number of integers within the specified range that are divisible by at least one of the four given numbers. While more advanced techniques exist, the inclusion-exclusion method remains the most accessible and effective approach for this specific scenario.