Understanding and Calculating Positive Integers Divisible by 1, 2, 3, 4, and 5

When tackling the problem of finding positive integers less than 600 that are divisible by 1, 2, 3, 4, and 5, the first step is to understand the concept of the Least Common Multiple (LCM). This article will walk you through the process of finding such numbers using both an analytical and a practical approach, ensuring that every step is clear and understandable.

Step 1: Determine the LCM

To find the LCM of the numbers 1, 2, 3, 4, and 5, we need to break down each number into its prime factors:

2 21

3 31

4 22

5 51

The LCM is found by taking the highest power of each prime factor that appears in the factorizations. For the numbers 1, 2, 3, 4, and 5, the LCM is calculated as follows:

Highest power of 2: 22

Highest power of 3: 31

Highest power of 5: 51

Therefore, the LCM is:

LCM(1, 2, 3, 4, 5) 22 × 31 × 51 4 × 3 × 5 60

Step 2: Counting Multiples of 60

With the LCM of 60, we can now count how many multiples of 60 are less than 600. Using the formula for counting multiples:

n ?(n-1) / k?

where k 60 and n 600, we can calculate:

?(600-1) / 60? ?599 / 60? ≈ ?9.9833? 9

Thus, there are 9 positive integers less than 600 that are divisible by all of the numbers 1, 2, 3, 4, and 5.

Conclusion

Breaking down the problem, we see:

All integers are divisible by 1, so this factor is taken care of. Since any number divisible by 4 is also divisible by 2, we only need to consider 3, 4, and 5. The LCM of 3, 4, and 5 is 60, and any multiple of 60 is divisible by 3, 4, and 5.

Therefore, the positive integers less than 600 that are divisible by 1, 2, 3, 4, and 5 are the multiples of 60, which are 60, 120, 180, 240, 300, 360, 420, 480, and 540. In total, there are 9 such numbers.

Additional Tips for Divisibility and LCM

Divisibility rules and the LCM are foundational concepts in number theory. Understanding these concepts can help in solving various mathematical problems, from finding common denominators to simplifying fractions. By mastering these techniques, you can approach similar problems more efficiently.

Keywords: Least Common Multiple (LCM), Divisibility, Positive Integers