Introduction to Finding the Least Common Multiple (LCM)

Understanding how to find the smallest number that has 12, 13, and 30 as factors is an essential skill in mathematics. This process involves finding the least common multiple (LCM) of these numbers. The LCM is the smallest number that is a multiple of each of the given numbers. In this article, we will explore two methods to find the LCM: prime factorization and the brute force method.

Prime Factorization Method for Finding the LCM

The prime factorization method is a straightforward and systematic approach to finding the LCM. First, let's perform the prime factorization of each number:

Step-by-Step Prime Factorization

To find the LCM of 12, 13, and 30, we begin by expressing each number in terms of its prime factors:

12 22 × 31

13 131

30 21 × 31 × 51

Determining the Highest Powers of Each Prime

Next, we identify the highest power of each prime number present in the factorizations:

The highest power of 2 is 22 from 12. The highest power of 3 is 31 from both 12 and 30. The highest power of 5 is 51 from 30. The highest power of 13 is 131 from 13.

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

LCM 22 × 31 × 51 × 131

Calculation of the LCM

Following the calculation:

22 4 31 3 51 5 131 13

Combining these values:

4 × 3 12 12 × 5 60 60 × 13 780

Hence, the smallest number that has 12, 13, and 30 as factors is 780, which is their least common multiple (LCM).

Brute-Force Method for Finding the LCM

While the prime factorization method is efficient, there is another approach that involves a brute-force method. This method is particularly useful when the given numbers are relatively small. Here’s how it works:

Start with the largest of the given numbers. Generate a list of multiples of this number. Check if each multiple is divisible by all the other given numbers. Continue this process until you find a number that satisfies the condition.

Let's illustrate this method with the numbers 12, 13, and 30:

Starting with 30:

30 is not divisible by 12. 60 is divisible by 12, but not by 13. 90 is not divisible by 13. 120 is divisible by 12, but not by 13. Continue checking until you find 780, which is divisible by all three numbers.

Conclusion and Summary

In conclusion, the smallest number that has 12, 13, and 30 as factors is 780, which can be determined using either the prime factorization method or the brute-force method. These methods provide a systematic way to find the least common multiple (LCM) of any set of numbers, making it a valuable skill in mathematical problem-solving.