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What is the Least Common Multiple using Common Division?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all those numbers. Using the common division method, we find the LCM by dividing the numbers together by their common prime factors until no more common factors exist.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya. Rahul gets pocket money every 3 days, and Priya gets pocket money every 5 days. If they both got pocket money today, on which day will they both get pocket money again at the same time? We need to find the LCM of 3 and 5, which is 15. So, they will both get pocket money together again after 15 days.
Worked Example
Step-by-Step
Let's find the LCM of 12 and 18 using the common division method.
Step 1: Write the numbers side-by-side: 12, 18.
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Step 2: Find the smallest prime number that divides at least one of them. Here, 2 divides both 12 and 18.
Divide both numbers by 2:
2 | 12, 18
| 6, 9
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Step 3: Look at the new numbers (6, 9). Again, find the smallest prime number that divides at least one. Here, 2 divides 6 but not 9. So, divide 6 by 2 and bring 9 down as it is:
2 | 6, 9
| 3, 9
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Step 4: Look at the new numbers (3, 9). The smallest prime number that divides at least one is 3. Divide both 3 and 9 by 3:
3 | 3, 9
| 1, 3
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Step 5: Look at the new numbers (1, 3). The smallest prime number that divides at least one is 3. Divide 3 by 3 and bring 1 down as it is:
3 | 1, 3
| 1, 1
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Step 6: Stop when all numbers become 1.
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Step 7: Multiply all the prime divisors you used: 2 x 2 x 3 x 3.
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Step 8: Calculate the product: 2 x 2 = 4, 4 x 3 = 12, 12 x 3 = 36.
So, the LCM of 12 and 18 is 36.
Why It Matters
Understanding LCM is super important in many fields! Engineers use it when designing gears or scheduling traffic lights to work smoothly. In computer science, it helps in optimizing processes and managing data efficiently. Even scientists use it to understand cycles and patterns in nature.
Common Mistakes
MISTAKE: Stopping division too early, when only common factors are found. | CORRECTION: Continue dividing by prime factors until all numbers become 1, even if a factor divides only one of the numbers.
MISTAKE: Forgetting to multiply all the prime factors used in the division. | CORRECTION: Make sure to multiply ALL the prime numbers listed on the left side of your division, including any numbers carried down that were not divided.
MISTAKE: Using composite numbers (like 4 or 6) as divisors instead of prime numbers. | CORRECTION: Always use only prime numbers (2, 3, 5, 7, etc.) for division in this method.
Practice Questions
Try It Yourself
QUESTION: Find the LCM of 8 and 10 using the common division method. | ANSWER: 40
QUESTION: What is the LCM of 6, 9, and 12? | ANSWER: 36
QUESTION: Three bells ring at intervals of 5, 10, and 15 minutes respectively. If they all ring together at 10:00 AM, when will they next ring together? | ANSWER: 10:30 AM (LCM of 5, 10, 15 is 30, so they ring together after 30 minutes)
MCQ
Quick Quiz
What is the LCM of 7 and 11?
1
7
11
77
The Correct Answer Is:
D
Since 7 and 11 are both prime numbers, their only common factor is 1. The LCM of two prime numbers is simply their product, so 7 x 11 = 77.
Real World Connection
In the Real World
Think about traffic lights at a busy junction in Delhi or Mumbai. If one light turns green every 40 seconds and another every 60 seconds, traffic engineers use LCM to figure out when both lights will turn green at the exact same moment again. This helps them synchronize signals to manage traffic flow efficiently and reduce jams.
Key Vocabulary
Key Terms
MULTIPLE: A number that can be divided by another number a certain number of times without a remainder. | PRIME NUMBER: A whole number greater than 1 that has exactly two positive divisors: 1 and itself. | COMMON FACTOR: A number that divides two or more other numbers exactly. | DIVISION METHOD: A systematic way of finding the LCM by repeatedly dividing numbers by prime factors. | LEAST COMMON MULTIPLE (LCM): The smallest positive number that is a multiple of two or more given numbers.
What's Next
What to Learn Next
Great job learning LCM! Now that you know how to find the smallest common multiple, you're ready to explore 'Highest Common Factor (HCF)'. HCF is another important concept that uses similar division skills and helps you understand relationships between numbers even better.
