Lcm Of 144 180 And 384 By Division Method

Finding the LCM of 144, 180, and 384 Using the Division Method

Finding the least common multiple (LCM) of numbers is a fundamental concept in mathematics with applications spanning various fields, from scheduling tasks to simplifying fractions. While calculators and online tools readily provide the LCM, understanding the underlying process—especially the division method—offers valuable insight into number theory and problem-solving. This comprehensive guide will delve into calculating the LCM of 144, 180, and 384 using the division method, explaining each step in detail and highlighting the underlying principles.

Understanding the Least Common Multiple (LCM)

Before we dive into the calculation, let's solidify our understanding of the LCM. The LCM of two or more integers is the smallest positive integer that is divisible by all the given integers. Think of it as the smallest common "multiple" that all the numbers share. For instance, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Several methods exist for calculating the LCM, including listing multiples, prime factorization, and the division method. We'll focus on the division method, which is particularly efficient when dealing with larger numbers like 144, 180, and 384.

The Division Method for Finding the LCM

The division method, also known as the prime factorization method, is a systematic approach that leverages the prime factors of the given numbers to find their LCM. It involves repeatedly dividing the numbers by their common prime factors until all the resulting quotients are 1. The LCM is then calculated by multiplying all the prime factors used in the process.

Steps involved in the division method:

1. Arrange the numbers: Write down the numbers whose LCM you want to find in a row. In our case, we have 144, 180, and 384.

2. Find a common prime factor: Identify the smallest prime number (2, 3, 5, 7, etc.) that divides at least one of the given numbers.

3. Divide: Divide each number by the chosen prime factor. If a number is not divisible by the prime factor, simply carry it down to the next row.

4. Repeat: Repeat steps 2 and 3 until all the numbers are reduced to 1.

5. Calculate the LCM: Multiply all the prime factors used in the division process to obtain the LCM.

Calculating the LCM of 144, 180, and 384

Let's apply the division method to find the LCM of 144, 180, and 384:

Step	144	180	384	Prime Factor
1	144	180	384
2	72	90	192	2
3	36	45	96	2
4	18	45	48	2
5	9	45	24	2
6	9	45	12	2
7	9	45	6	2
8	9	45	3	2
9	3	15	1	3
10	1	5	1	3
11	1	1	1	5

Explanation of the steps:

• Step 1: We start with the numbers 144, 180, and 384.

• Steps 2-8: We repeatedly divide by 2 until no number is divisible by 2 anymore. Notice that we continue to divide even if only one of the numbers is divisible.

• Step 9: We move to the next prime number, 3. We divide by 3.

• Step 10: We move to the next prime number, 5 and divide.

• Step 11: All numbers have been reduced to 1.

Calculating the LCM:

To obtain the LCM, we multiply all the prime factors used in the division process:

LCM(144, 180, 384) = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2⁶ × 3² × 5 = 64 × 9 × 5 = 2880

Therefore, the least common multiple of 144, 180, and 384 is 2880.

Verification and Applications

To verify our result, we can check if 2880 is divisible by 144, 180, and 384:

• 2880 ÷ 144 = 20
• 2880 ÷ 180 = 16
• 2880 ÷ 384 = 7.5 (This is incorrect, let's review the steps)

Reviewing the Calculations: There was a mistake made in Step 8: 9 is not divisible by 2. Let's correct it.

Step	144	180	384	Prime Factor
1	144	180	384
2	72	90	192	2
3	36	45	96	2
4	18	45	48	2
5	9	45	24	2
6	9	45	12	2
7	9	45	6	2
8	9	45	3	2
9	3	15	1	3
10	1	5	1	3
11	1	5	1	3
12	1	1	1	5

Now let's recalculate:

LCM(144, 180, 384) = 2⁶ x 3² x 5 = 64 x 9 x 5 = 2880

Now the divisions are correct. 2880/144 = 20, 2880/180 = 16, 2880/384 = 7.5. There must be another error. Let's double-check the prime factorization:

144 = 2⁴ x 3² 180 = 2² x 3² x 5 384 = 2⁷ x 3

To find the LCM, we take the highest power of each prime factor present: 2⁷ x 3² x 5 = 128 x 9 x 5 = 5760

Therefore, the correct LCM of 144, 180, and 384 is 5760.

Real-world Applications of LCM

Understanding and calculating the LCM has practical applications in various scenarios:

• Scheduling: Imagine you have three machines that complete a cycle in 144, 180, and 384 minutes, respectively. The LCM (5760 minutes) determines when all three machines will complete a cycle simultaneously.

• Fractions: Finding the LCM of denominators is crucial when adding or subtracting fractions. It helps in finding the least common denominator (LCD).

• Project Management: In project planning, the LCM can be used to synchronize tasks or activities that have different durations.

• Music: LCM plays a role in musical harmony and rhythm, determining when different musical phrases will align.

Conclusion

Calculating the LCM using the division method offers a structured and efficient way to find the least common multiple of multiple numbers. While seemingly a simple mathematical concept, understanding LCM is essential for various applications across diverse fields. Through careful application of the division method and double-checking our work, we've accurately determined that the LCM of 144, 180, and 384 is 5760. Remember to always carefully check your calculations to avoid errors like the one encountered initially. This comprehensive guide provides a clear understanding of the process and highlights the importance of this mathematical concept.