Understanding And Solving Lcm Of 30 And 45

LCM of 30 and 45 How to Find LCM of 30, 45? from www.cuemath.com

What is LCM?

LCM stands for Least Common Multiple which refers to the smallest number that is a multiple of two or more given numbers. It is also known as the Lowest Common Multiple.

Prime Factorization of 30 and 45

To find the LCM of 30 and 45, we need to first determine the prime factorization of both numbers. Prime factorization is the process of finding the prime factors of a number. Prime factors are numbers that are only divisible by 1 and themselves. To find the prime factorization of 30, we need to divide it by its smallest prime factor, which is 2. 30 ÷ 2 = 15 15 ÷ 3 = 5 Therefore, the prime factorization of 30 is 2 x 3 x 5. To find the prime factorization of 45, we also divide it by its smallest prime factor, which is 3. 45 ÷ 3 = 15 15 ÷ 3 = 5 Therefore, the prime factorization of 45 is 3 x 3 x 5.

Finding the LCM

To find the LCM of 30 and 45, we need to identify the highest power of each prime factor that appears in either of the two factorizations. We can see that both factorizations contain 3 and 5 as common prime factors. The highest power of 2 that appears in the factorization of 30 is 2^1, while the factorization of 45 does not contain 2 as a prime factor. Therefore, we need to include 2^1 in the LCM. The highest power of 3 that appears in the factorization of 30 is 3^1, while the factorization of 45 contains 3^2. Therefore, we need to include 3^2 in the LCM. Therefore, the LCM of 30 and 45 is 2 x 3^2 x 5, which simplifies to 90.

Why is LCM Important?

LCM is an important concept in mathematics as it is used to find solutions to real-world problems. For example, when planning a school trip, the LCM can be used to determine the least number of days that all students can go on the trip.

Other Methods of Finding LCM

Apart from finding the prime factorization, there are other methods to find the LCM of two or more numbers. One such method is the Prime Factorization Method, where we list down the prime factors of each number and then multiply the highest powers of each factor. Another method is the Division Method, where we divide the numbers by their highest common factor, and then multiply the quotient by the common factors and the remaining factors.

Practice Questions

1. Find the LCM of 15 and 20. 2. Find the LCM of 24, 36, and 48.

Conclusion

The LCM of 30 and 45 is 90, which is the smallest number that is a multiple of both 30 and 45. LCM is an important concept in mathematics that is used to solve various problems. By understanding and practicing the different methods to find LCM, you can improve your mathematical skills and problem-solving abilities.