Introduction

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In this article, we will explore the process of counting positive integers less than 600 that are divisible by 3 or 7. This problem requires the application of the Inclusion-Exclusion Principle, a fundamental concept in number theory and set theory. By breaking down the problem into simpler parts and using the principle, we can find the total count efficiently. Let's delve into the details.

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Step-by-Step Solution

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To solve this problem, we will follow a systematic approach that involves counting the multiples of 3, the multiples of 7, and then adjusting for the overlap.

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Counting Multiples of 3

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The first step is to count how many positive integers less than 600 are divisible by 3. The sequence of numbers divisible by 3 forms an arithmetic sequence: 3, 6, 9, …, 597.

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Mathematically, we can determine the number of terms in this sequence using the formula for finding the number of terms in an arithmetic sequence:

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n lfloor 599 / 3 rfloor

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By performing the division:

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599 / 3 asymp; 199.666….

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Therefore, the number of positive integers less than 600 that are divisible by 3 is:

r r r n 199r r r

Counting Multiples of 7

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Next, we count the multiples of 7. The sequence of numbers divisible by 7 is: 7, 14, 21, …, 595.

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Similarly, we can determine the number of terms in this sequence using the same formula:

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n lfloor 599 / 7 rfloor

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By performing the division:

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599 / 7 asymp; 85.571…

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Therefore, the number of positive integers less than 600 that are divisible by 7 is:

r r r n 85r r r

Counting Multiples of Both 3 and 7 (i.e., 21)

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Some numbers are divisible by both 3 and 7. These numbers are the multiples of their least common multiple (LCM), which is 21. The sequence of numbers divisible by 21 is: 21, 42, 63, …, 588.

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Again, we can determine the number of terms in this sequence using the same formula:

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n lfloor 599 / 21 rfloor

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By performing the division:

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599 / 21 asymp; 28.523…

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Therefore, the number of positive integers less than 600 that are divisible by 21 is:

r r r n 28r r r

Applying the Inclusion-Exclusion Principle

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According to the Inclusion-Exclusion Principle, the total number of positive integers less than 600 that are divisible by 3 or 7 is given by:

r r r Total (Multiples of 3) (Multiples of 7) - (Multiples of both 3 and 7)r r r

Substituting the values:

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Total 199 85 - 28 256

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Therefore, the total number of positive integers less than 600 that are divisible by 3 or 7 is:

r r r 256r r r

Conclusion

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In this article, we have explored the mathematical method to find the total count of positive integers less than 600 that are divisible by 3 or 7. This involves a systematic approach using the Inclusion-Exclusion Principle. Understanding and applying this principle can help solve similar counting problems in number theory and related mathematical fields.