What is the smallest positive integer that has 6, 8, and 10 as its factors?

To find the smallest positive integer that has 6, 8, and 10 as its factors, we need to determine the least common multiple (LCM) of these three numbers. The LCM is the smallest number that is a multiple of all the given numbers.

Prime Factorization Method

First, we find the prime factorization of each number:

6: 21 × 31 8: 23 10: 21 × 51

Next, we identify the highest power of each prime factor that appears in the factorizations:

2: The highest power is 23 from 8. 3: The highest power is 31 from 6. 5: The highest power is 51 from 10.

Now we can calculate the LCM by multiplying these highest powers together:

LCM 23 × 31 × 51

Step-by-Step Calculation

Let's calculate it step-by-step:

23 8 31 3 51 5

Now multiply these together:

8 × 3 24

24 × 5 120

Therefore, the least common multiple of 6, 8, and 10 is 120. Hence, the smallest positive integer that has 6, 8, and 10 as its factors is 120.

Brute Force Method

A brute force approach to finding the LCM involves iterating through numbers starting from 120 (since we know the LCM must be at least 120) and checking if it is divisible by 6, 8, and 10. Here’s a basic implementation in C:

#include stdio.hint main() {    int n  120;    while (n % 6 ! 0 || n % 8 ! 0 || n % 10 ! 0) {        n  ;    }    printf("%d
", n);  // Output: 120    return 0;}

This program confirms that 120 is the smallest number that has 6, 8, and 10 as its factors.

Conclusion

In conclusion, the least common multiple (LCM) of 6, 8, and 10 is 120. This can be determined using prime factorization or a brute force approach.

Key Takeaways:

The LCM of 6, 8, and 10 is 120. Prime factorization helps in determining the LCM by identifying the highest powers of each prime factor. A brute force method can also verify the LCM.

Related Topics

LCM, Prime Factorization, and Number Theory Basics

Understanding LCM and prime factorization is essential in number theory and is used in various fields such as computer science, cryptography, and engineering. This knowledge can help in solving more complex problems involving multiples and factors.