What is Long Division Method for LCM?

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Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Long Division Method for LCM is a simple way to find the Least Common Multiple (LCM) of two or more numbers. In this method, we divide the given numbers by their common prime factors until we can't divide them further. The LCM is then found by multiplying all the divisors and the remaining undivided numbers.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya. Rahul visits the library every 4 days, and Priya visits every 6 days. If they both visit today, when will they visit the library together again? To find this, we need the LCM of 4 and 6, which we can find using the Long Division Method.

Worked Example

Step-by-Step

Let's find the LCM of 12, 18, and 24 using the Long Division Method.

Step 1: Write the numbers in a row: 12, 18, 24.

Step 2: Start dividing by the smallest prime number (2) that divides at least two of the numbers.
2 | 12, 18, 24
| 6, 9, 12

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Step 3: Again, divide by 2. (9 is not divisible by 2, so we bring it down as is).
2 | 6, 9, 12
| 3, 9, 6

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Step 4: Again, divide by 2. (3 and 9 are not divisible by 2, so we bring them down).
2 | 3, 9, 6
| 3, 9, 3

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Step 5: Now, no number is divisible by 2. Move to the next smallest prime number (3).
3 | 3, 9, 3
| 1, 3, 1

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Step 6: Again, divide by 3. (1s are brought down).
3 | 1, 3, 1
| 1, 1, 1

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Step 7: Stop when all the numbers in the last row are 1.

Step 8: Multiply all the prime divisors: 2 x 2 x 2 x 3 x 3 = 72.

So, the LCM of 12, 18, and 24 is 72.

Why It Matters

Understanding LCM is super important! In computer science, it helps in scheduling tasks or optimizing algorithms. Engineers use it to design gears or synchronize cycles. Even in data science, it can be used for pattern recognition. Learning this skill can open doors to exciting careers as a Software Developer or a Robotics Engineer!

Common Mistakes

MISTAKE: Not using only prime numbers as divisors. | CORRECTION: Always divide by prime numbers like 2, 3, 5, 7, etc. Never use composite numbers like 4, 6, 8.

MISTAKE: Forgetting to bring down numbers that are not divisible by the current divisor. | CORRECTION: If a number is not perfectly divisible, write it as it is in the next row.

MISTAKE: Stopping too early, before all numbers in the last row become 1. | CORRECTION: Continue dividing until all numbers in the final row are 1. Only then can you multiply all divisors.

Practice Questions

Try It Yourself

QUESTION: Find the LCM of 8 and 12 using the Long Division Method. | ANSWER: 24

QUESTION: What is the LCM of 10, 15, and 20? | ANSWER: 60

QUESTION: Three traffic lights at a crossing change every 30 seconds, 45 seconds, and 60 seconds respectively. If they all change simultaneously at 8:00 AM, when will they next change together? Find the LCM of 30, 45, and 60. | ANSWER: 180 seconds (or 3 minutes), so they will next change together at 8:03 AM.

MCQ

Quick Quiz

What is the first step in finding the LCM of 6 and 9 using the Long Division Method?

Multiply 6 and 9

Divide both numbers by their smallest common prime factor

List multiples of 6 and 9

Add 6 and 9

The Correct Answer Is:

The first step in the Long Division Method for LCM is to divide the given numbers by their smallest common prime factor. Options A, C, and D are incorrect steps for this method.

Real World Connection

In the Real World

Think about planning a big family trip where everyone needs to meet at a railway station. If trains from different cities arrive every 3 hours, 4 hours, and 6 hours, finding the LCM helps you figure out the earliest time all family members can meet if they start at the same time. This ensures no one waits too long!

Key Vocabulary

Key Terms

LCM: Least Common Multiple, the smallest number that is a multiple of two or more numbers. | Prime Factor: A prime number that divides another number exactly. | Divisor: A number that divides another number exactly without leaving a remainder. | Multiple: A number that can be divided by another number a certain number of times without a remainder.

What's Next

What to Learn Next

Great job learning about LCM! Next, you can explore 'What is the HCF (Highest Common Factor) using the Prime Factorization Method?' This will help you understand more about factors and multiples, which are basic building blocks for many advanced math concepts.