Understanding The Lcm Of 24 And 30

LCM of 24 and 30 How to Find LCM of 24, 30? from www.cuemath.com

What is LCM?

LCM stands for "Least Common Multiple." It is the smallest number that is a multiple of two or more given numbers. In other words, it is the smallest number that is divisible by all the given numbers without any remainder.

Factors of 24 and 30

To find the LCM of 24 and 30, we need to first list down the factors of both numbers. Factors are the numbers that can divide a given number without leaving a remainder. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Method 1: Prime Factorization

One way to find the LCM of 24 and 30 is by using prime factorization. Prime factorization means expressing a number as a product of its prime factors. Prime factors are the prime numbers that can divide a given number without leaving a remainder. To do this, we first list down the prime factors of both numbers. The prime factors of 24 are 2, 2, 2, and 3. The prime factors of 30 are 2, 3, and 5. We then find the highest power of each prime factor that appears in both lists. In this case, the highest power of 2 is 2^3, the highest power of 3 is 3^1, and the highest power of 5 is 5^1. We then multiply these numbers together to get the LCM, which is 2^3 x 3^1 x 5^1 = 120.

Method 2: Listing Multiples

Another way to find the LCM of 24 and 30 is by listing the multiples of the larger number until we find a multiple that is divisible by the smaller number. We start by listing the multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200. We can see that 120 is divisible by 24, so the LCM is 120.

Method 3: Division Method

A third way to find the LCM of 24 and 30 is by using the division method. We start by dividing one of the numbers by their common factors until we get a quotient that is not divisible by any of the common factors. We then multiply the quotient by the common factors and repeat the process until we get the LCM. Let's start with 24. We can divide it by 2, which gives us 12. 12 is also divisible by 2, so we divide it again and get 6. 6 is divisible by 2 and 3, so we divide it by 2 and get 3. 3 is not divisible by any of the common factors, so we multiply it by 2 x 2 x 2 x 3 = 24. We then repeat the process with 30. We can divide it by 2 and get 15. 15 is not divisible by 2, but it is divisible by 3 and 5. We divide it by 3 and get 5. 5 is not divisible by any of the common factors, so we multiply it by 2 x 2 x 2 x 3 x 5 = 120. Therefore, the LCM of 24 and 30 is 120.

Conclusion

In conclusion, there are different ways to find the LCM of 24 and 30. We can use prime factorization, listing multiples, or the division method. Regardless of the method we use, the answer will always be the same. The LCM of 24 and 30 is 120.