What is the 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. It involves dividing the numbers by common prime factors until no two numbers share a common factor, then multiplying all the divisors and remaining numbers.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya. Rohan visits the library every 4 days, and Priya visits every 6 days. To find out when they will meet at the library next, you need to find the LCM of 4 and 6. Using the long division method, we'd find their LCM is 12, meaning they'll meet again in 12 days.

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

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Step 2: Find the smallest prime number that divides at least two of the numbers. Here, 2 divides all three.
2 | 12, 18, 24
| 6, 9, 12

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Step 3: Repeat the process with the new row of numbers. Again, 2 divides 6 and 12.
2 | 12, 18, 24
2 | 6, 9, 12
| 3, 9, 6

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Step 4: Repeat. 3 divides 3, 9, and 6.
2 | 12, 18, 24
2 | 6, 9, 12
3 | 3, 9, 6
| 1, 3, 2

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Step 5: No two numbers in the last row (1, 3, 2) share a common prime factor other than 1. So, we stop.

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Step 6: Multiply all the divisors and the remaining numbers in the last row.
LCM = 2 x 2 x 3 x 1 x 3 x 2

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Step 7: Calculate the product.
LCM = 4 x 3 x 6
LCM = 12 x 6
LCM = 72

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

Why It Matters

Understanding LCM is super important in many advanced fields! Engineers use it to design gears and circuits that work together smoothly. Computer scientists use it in algorithms for scheduling tasks efficiently, and even data scientists might use it for pattern recognition in large datasets.

Common Mistakes

MISTAKE: Stopping division too early, even if two numbers still share a common prime factor. | CORRECTION: Continue dividing until no two numbers in the row can be divided by the same prime number.

MISTAKE: Forgetting to bring down numbers that are not divisible by the chosen prime factor. | CORRECTION: If a number is not divisible, simply write it as it is in the next row.

MISTAKE: Only multiplying the common prime factors and not the remaining numbers in the last row. | CORRECTION: The LCM is the product of ALL the divisors AND all the numbers left in the final row.

Practice Questions

Try It Yourself

QUESTION: Find the LCM of 8 and 10 using the long division method. | ANSWER: 40

QUESTION: What is the LCM of 15, 20, and 25? | ANSWER: 300

QUESTION: Three traffic lights at a busy Mumbai intersection change every 30 seconds, 45 seconds, and 60 seconds respectively. If they all change at the same time right now, after how many seconds will they all change together again? | ANSWER: 180 seconds

MCQ

Quick Quiz

Which of these is the correct first step when finding the LCM of 9, 15, and 21 using the long division method?

Divide all numbers by 2

Divide all numbers by 3

Divide 9 by 3 and 15 by 5

Multiply 9, 15, and 21

The Correct Answer Is:

The first step is to find the smallest prime number that divides at least two of the numbers. Here, 3 divides 9, 15, and 21. Options A, C, and D are incorrect or not the most efficient first step.

Real World Connection

In the Real World

Imagine you're a railway engineer planning train schedules. If one train leaves a station every 3 hours and another every 4 hours, you'd use LCM to find out when they'll both depart at the same time again, helping to avoid congestion and plan maintenance. This helps keep India's vast railway network running smoothly!

Key Vocabulary

Key Terms

LCM: Least Common Multiple, the smallest number that is a multiple of two or more given 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 mastering the Long Division Method for LCM! Next, you can explore the relationship between LCM and HCF (Highest Common Factor). Understanding both will help you solve even more complex problems in fractions and ratios.