To find the Least Common Multiple (LCM) of 10 and 12, we first need to understand what LCM is. The LCM of two numbers is the smallest number that is a multiple of both. Let’s break it down into simple steps:
List the Multiples: First, we list the multiples of each number.
- Multiples of 10: 10, 20, 30, 40, 50, 60…
- Multiples of 12: 12, 24, 36, 48, 60…
Identify the Smallest Common Multiple: From the lists, we identify the smallest number that appears in both lists.
- The first number that appears in both lists is 60.
Therefore, the LCM of 10 and 12 is 60.
Alternative Method: Using Prime Factorization
Another way to find the LCM is by using prime factorization. This method involves breaking down each number into its prime factors and then taking the highest power of all prime factors involved.
Prime Factorize Each Number:
- The prime factorization of 10 is: 2 * 5
- The prime factorization of 12 is: 2^2 * 3
Take the Highest Power of Each Prime Factor:
- For the prime factor 2, the highest power between the two numbers is 2^2 (from 12).
- For the prime factor 3, it only appears in 12, so we take 3.
- For the prime factor 5, it only appears in 10, so we take 5.
Multiply These Prime Factors:
- LCM = 2^2 * 3 * 5 = 4 * 3 * 5 = 60
Thus, using prime factorization, we also find that the LCM of 10 and 12 is 60.
Conclusion
Finding the LCM of two numbers can be achieved through simple methods like listing multiples or more systematic approaches like prime factorization. Both methods lead to the same result, which is essential for various mathematical operations and real-world applications. In this case, the LCM of 10 and 12 is 60, which means that 60 is the smallest number that both 10 and 12 can divide into evenly.
