Understanding And Solving For The Lcm Of 15 And 12

LCM of 12 and 15 How to Find LCM of 12, 15? from www.cuemath.com

Introduction

As a student, one of the most common mathematical problems you will encounter is finding the Least Common Multiple (LCM) of two numbers. The LCM is the smallest positive integer that is divisible by two or more numbers without leaving a remainder. In this article, we will focus on understanding and solving for the LCM of 15 and 12.

What are Multiples?

A multiple is a product of a given number and any other whole number. For example, the multiples of 15 are 15, 30, 45, 60, and so on. Similarly, the multiples of 12 are 12, 24, 36, 48, and so on.

What is the LCM?

The LCM is the smallest common multiple of the given numbers. In other words, it is the smallest number that both 15 and 12 divide into without leaving a remainder. The LCM of 15 and 12 can be found by listing their multiples and finding the smallest one that they have in common.

Listing Multiples

To find the LCM of 15 and 12, we first list their multiples. The multiples of 15 are 15, 30, 45, 60, 75, 90, and so on. The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, and so on.

Identifying the Common Multiple

From the list of multiples, we can see that 60 is the smallest number that both 15 and 12 divide into without leaving a remainder. Therefore, 60 is the LCM of 15 and 12.

Using Prime Factorization

Another method of finding the LCM of two numbers is by using prime factorization. To use this method, we first find the prime factors of each number. The prime factors of 15 are 3 and 5, while the prime factors of 12 are 2, 2, and 3.

Multiplying Prime Factors

Next, we multiply each prime factor the greatest number of times it occurs in either of the two numbers. In this case, the greatest number of times 2 occurs is twice in 12, while the greatest number of times 3 and 5 occur is once in 15. Therefore, the LCM of 15 and 12 is 2 x 2 x 3 x 5 = 60.

The Importance of LCM

The LCM is an important concept in mathematics because it is used in many different fields, such as engineering, physics, and computer science. For example, when computing the time it takes for two people to complete a task working together, the LCM of their individual work rates is used.

Conclusion

To sum up, the LCM of 15 and 12 is 60. It can be found by listing the multiples of both numbers and identifying the smallest common multiple or by using prime factorization. The LCM is an important concept in mathematics that is used in many different fields.