Positive Integers Multiples Of 9, 4, Or 7 Below 1475 Using Inclusion-Exclusion

Jul 10, 2025 by ADMIN 79 views

How to Find Multiples of 9, 4, or 7 Below 1475 Using the Principle of Inclusion-Exclusion

In the realm of number theory, a fascinating question arises: how many positive integers less than 1475 are multiples of 9, 4, or 7? This problem, seemingly straightforward, unveils the elegance and power of the Principle of Inclusion-Exclusion (PIE). This principle provides a systematic approach to counting elements in the union of multiple sets, especially when those sets overlap. In this comprehensive guide, we will embark on a journey to dissect this problem, unravel the intricacies of PIE, and ultimately arrive at a solution. Prepare to delve into the world of multiples, sets, and the beauty of mathematical reasoning.

Understanding the Problem: Multiples and the Range

To effectively tackle this problem, we must first grasp the fundamental concepts involved. Multiples of a number are the results obtained by multiplying that number by any integer. For instance, the multiples of 9 are 9, 18, 27, 36, and so on. Similarly, multiples of 4 are 4, 8, 12, 16, and so forth, while multiples of 7 are 7, 14, 21, 28, and so on. Our focus is on positive integers, meaning we only consider whole numbers greater than zero.

Our range of interest is defined as less than 1475. This means we are looking for multiples that fall within the numbers 1 to 1474. This limitation is crucial because it provides an upper bound, allowing us to determine a finite set of multiples for each number (9, 4, and 7). Without this constraint, the problem would become infinite, as there would be an unlimited number of multiples.

The core challenge lies in the fact that some numbers are multiples of more than one of our target numbers (9, 4, and 7). For example, 36 is a multiple of both 9 and 4. Simply counting the multiples of each number separately and adding them together would lead to overcounting these common multiples. This is where the Principle of Inclusion-Exclusion comes into play, providing a method to correct for this overcounting and arrive at the accurate answer.

Therefore, understanding the concept of multiples, the given range, and the potential for overlap is the cornerstone for solving this problem. With these fundamentals in place, we can now move on to exploring the Principle of Inclusion-Exclusion in greater detail.

The Principle of Inclusion-Exclusion: A Detailed Explanation

The Principle of Inclusion-Exclusion (PIE) is a powerful counting technique used to determine the number of elements in the union of multiple sets. It's particularly useful when these sets have overlaps, meaning some elements belong to more than one set. The basic idea behind PIE is to first include the sizes of each individual set, then exclude the sizes of the intersections of pairs of sets, then include the sizes of the intersections of triplets of sets, and so on, alternating between inclusion and exclusion until all possible intersections have been considered.

To illustrate this with our problem, let's define our sets:

Let A be the set of positive integers less than 1475 that are multiples of 9.
Let B be the set of positive integers less than 1475 that are multiples of 4.
Let C be the set of positive integers less than 1475 that are multiples of 7.

Our goal is to find the number of elements in the union of these three sets, denoted as |A ∪ B ∪ C|, which represents the total number of positive integers less than 1475 that are multiples of 9, 4, or 7.

The PIE formula for three sets is as follows:

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

Let's break down each term in the formula:

|A|, |B|, |C|: These represent the number of elements in each individual set. In our context, this is the number of multiples of 9, 4, and 7, respectively, that are less than 1475.
|A ∩ B|, |A ∩ C|, |B ∩ C|: These represent the number of elements in the intersections of pairs of sets. For example, |A ∩ B| is the number of integers that are multiples of both 9 and 4 (i.e., multiples of their least common multiple). We subtract these because we've initially counted them twice (once in |A| and once in |B|).
|A ∩ B ∩ C|: This represents the number of elements in the intersection of all three sets. In our case, it's the number of integers that are multiples of 9, 4, and 7. We add this back because we initially included it three times (in |A|, |B|, and |C|), then subtracted it three times (in |A ∩ B|, |A ∩ C|, and |B ∩ C|). Thus, it needs to be included once to correct the count.

In essence, the Principle of Inclusion-Exclusion ensures that we count each element exactly once by systematically including and excluding the appropriate overlaps. This principle is not limited to three sets; it can be extended to any number of sets by continuing the alternating pattern of inclusion and exclusion.

With a clear understanding of the Principle of Inclusion-Exclusion, we are now equipped to apply it to our specific problem and calculate the number of positive integers less than 1475 that are multiples of 9, 4, or 7.

Applying PIE to the Problem: Calculations and Solutions

Now that we understand the Principle of Inclusion-Exclusion (PIE), we can apply it to our problem. Recall that we need to find the number of positive integers less than 1475 that are multiples of 9, 4, or 7. We've already defined our sets A, B, and C, and we know the PIE formula for three sets:

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

The next step is to calculate each term in the formula. This involves determining the number of multiples within our specified range (less than 1475).

1. Calculating |A|, |B|, and |C|

|A| (Multiples of 9): To find the number of multiples of 9 less than 1475, we divide 1474 (the largest integer less than 1475) by 9 and take the integer part of the result: |A| = ⌊1474 / 9⌋ = 163 So, there are 163 multiples of 9 less than 1475.
|B| (Multiples of 4): Similarly, we divide 1474 by 4: |B| = ⌊1474 / 4⌋ = 368 There are 368 multiples of 4 less than 1475.
|C| (Multiples of 7): Dividing 1474 by 7: |C| = ⌊1474 / 7⌋ = 210 There are 210 multiples of 7 less than 1475.

2. Calculating Intersections of Pairs: |A ∩ B|, |A ∩ C|, and |B ∩ C|

To find the number of elements in the intersection of two sets, we need to find the least common multiple (LCM) of the corresponding numbers.

|A ∩ B| (Multiples of 9 and 4): The LCM of 9 and 4 is 36. We divide 1474 by 36: |A ∩ B| = ⌊1474 / 36⌋ = 40 There are 40 multiples of both 9 and 4 (i.e., multiples of 36) less than 1475.
|A ∩ C| (Multiples of 9 and 7): The LCM of 9 and 7 is 63. We divide 1474 by 63: |A ∩ C| = ⌊1474 / 63⌋ = 23 There are 23 multiples of both 9 and 7 (i.e., multiples of 63) less than 1475.
|B ∩ C| (Multiples of 4 and 7): The LCM of 4 and 7 is 28. We divide 1474 by 28: |B ∩ C| = ⌊1474 / 28⌋ = 52 There are 52 multiples of both 4 and 7 (i.e., multiples of 28) less than 1475.

3. Calculating the Intersection of All Three Sets: |A ∩ B ∩ C|

To find the number of elements in the intersection of all three sets, we need to find the LCM of 9, 4, and 7. The LCM of 9, 4, and 7 is 252. We divide 1474 by 252:

|A ∩ B ∩ C| (Multiples of 9, 4, and 7): |A ∩ B ∩ C| = ⌊1474 / 252⌋ = 5 There are 5 multiples of 9, 4, and 7 (i.e., multiples of 252) less than 1475.

4. Applying the PIE Formula

Now we have all the components to plug into the PIE formula:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| |A ∪ B ∪ C| = 163 + 368 + 210 - 40 - 23 - 52 + 5 |A ∪ B ∪ C| = 621

Therefore, there are 621 positive integers less than 1475 that are multiples of 9, 4, or 7.

By systematically applying the Principle of Inclusion-Exclusion and performing the necessary calculations, we have successfully solved this problem. This demonstrates the power and utility of PIE in counting problems involving overlapping sets.

Conclusion: The Power of Inclusion-Exclusion

In conclusion, the problem of finding the number of positive integers less than 1475 that are multiples of 9, 4, or 7 beautifully illustrates the application and importance of the Principle of Inclusion-Exclusion (PIE). This principle provides a robust and systematic method for counting elements in the union of sets, especially when overlaps exist. By carefully including the sizes of individual sets, excluding the sizes of pairwise intersections, including the sizes of triple intersections, and so on, PIE ensures that each element is counted exactly once.

We began by understanding the problem, defining our sets (A, B, and C) as the multiples of 9, 4, and 7, respectively, within the range of 1 to 1474. We then delved into a detailed explanation of PIE, highlighting its core idea of alternating between inclusion and exclusion to correct for overcounting. The PIE formula for three sets, |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|, became our guiding equation.

Next, we meticulously calculated each term in the formula. This involved finding the number of multiples of each individual number (9, 4, and 7), as well as the number of multiples of their pairwise and triple combinations (using the least common multiple). The calculations demonstrated how PIE systematically accounts for the overlaps between the sets.

Finally, we plugged the calculated values into the PIE formula and arrived at the solution: there are 621 positive integers less than 1475 that are multiples of 9, 4, or 7. This result underscores the accuracy and efficiency of PIE in solving such counting problems.

The Principle of Inclusion-Exclusion is not just a mathematical formula; it's a valuable problem-solving tool applicable in various fields, including computer science, statistics, and even everyday situations where counting overlapping groups is necessary. Mastering PIE enhances our ability to think critically and approach complex counting problems with confidence. Its elegance lies in its simplicity and its ability to break down seemingly intricate problems into manageable steps. As we have seen, by understanding and applying PIE, we can unravel the mysteries of number theory and beyond.