In this article, we will answer the commonly asked question, “what is lcm of 10 and 12?” Learn how to find the least common multiple of these two numbers.
Introduction
The concept of Least Common Multiple (LCM) is an essential part of mathematics that is used to solve many problems. LCM is defined as the smallest positive integer that is a multiple of two or more numbers. It is a fundamental concept in arithmetic and is often used in algebra, number theory, and other areas of mathematics. In this article, we will discuss the LCM of 10 and 12, which is a commonly asked question in mathematics.
Definition of LCM
The LCM of two or more numbers is the smallest multiple that they have in common. For example, the LCM of 2 and 3 is 6, which is the smallest multiple that they have in common. The LCM of 10 and 12 is the smallest positive integer that is a multiple of both 10 and 12.
To find the LCM of 10 and 12, we need to list the multiples of each number and find the smallest multiple that they have in common. The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, and so on. The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, and so on.
We can see that the smallest multiple that they have in common is 60. Therefore, the LCM of 10 and 12 is 60. In other words, 60 is the smallest positive integer that is a multiple of both 10 and 12.
Using the formula to calculate LCM, we can write LCM(10, 12) = (10 x 12) / GCD(10, 12), where GCD stands for the Greatest Common Divisor. The GCD of 10 and 12 is 2, so we have LCM(10, 12) = (10 x 12) / 2 = 60.
The LCM of 10 and 12 can also be found using the prime factorization method. To find the prime factorization of 10, we can write 10 as 2 x 5. To find the prime factorization of 12, we can write 12 as 2 x 2 x 3. The LCM of 10 and 12 is the product of the highest powers of all the prime factors, which is 2 x 2 x 3 x 5 = 60.
Finding the factors of 10 and 12
Factors are the numbers that divide a given number without leaving any remainder. To find the factors of 10 and 12, we need to list all the numbers that divide them without leaving a remainder.
The factors of 10 are 1, 2, 5, and 10. We can see that 1 and 10 are the factors that are common to both 10 and 12.
The factors of 12 are 1, 2, 3, 4, 6, and 12. We can see that 2, 3, 4, 6, and 12 are the factors of 12.
Identifying the common factors
To find the LCM of two or more numbers, we need to identify the common factors between them. In the case of 10 and 12, we can see that the factors 1 and 2 are common to both numbers.
The common factors of 10 and 12 are 1 and 2. We can see that 1 is a common factor of all the numbers, while 2 is the only common prime factor between them.
Identifying the common factors is an essential step in finding the LCM of any two or more numbers. Once we have identified the common factors, we can use any of the methods mentioned above to find the LCM. In the case of 10 and 12, we can use the fact that the LCM is the smallest multiple that they have in common to find the LCM, which is 60.
Finding the LCM
To find the LCM of 10 and 12, we can use various methods. One of the simplest methods is to list the multiples of both numbers and find the smallest multiple that they have in common. Another method is to use the prime factorization method, as mentioned earlier. We can also use the formula LCM(a, b) = (a x b) / GCD(a, b), where a and b are the two numbers and GCD is the Greatest Common Divisor.
Steps to find the LCM of 10 and 12

List the multiples of both numbers: The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, and so on. The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, and so on.

Identify the smallest multiple that they have in common: From the list of multiples, we can see that the smallest multiple that they have in common is 60.

Therefore, the LCM of 10 and 12 is 60.

Alternatively, we can use the formula LCM(a, b) = (a x b) / GCD(a, b). The GCD of 10 and 12 is 2, so we have LCM(10, 12) = (10 x 12) / 2 = 60.

We can also use the prime factorization method to find the LCM of 10 and 12. The prime factorization of 10 is 2 x 5, and the prime factorization of 12 is 2 x 2 x 3. The LCM of 10 and 12 is the product of the highest powers of all the prime factors, which is 2 x 2 x 3 x 5 = 60.
Conclusion
In conclusion, the LCM of 10 and 12 is 60. We have discussed various methods to find the LCM of two numbers, including listing the multiples, using the prime factorization method, and using the formula LCM(a, b) = (a x b) / GCD(a, b). LCM is an essential concept in mathematics that is used in various fields, including algebra, number theory, and computer science. Knowing how to find the LCM of two or more numbers is crucial in solving many problems in mathematics.