Least Common Multiple Of 15 And 6 Explained Simply
Table of Contents :
To understand the least common multiple (LCM) of the numbers 15 and 6, let's break it down in simple terms. The LCM is the smallest positive integer that is divisible by both numbers. This means that the LCM of 15 and 6 is the smallest number into which both 15 and 6 can evenly divide without leaving a remainder. To get there, we can employ different methods: listing multiples, using the prime factorization approach, or utilizing the relationship between the greatest common divisor (GCD) and LCM. Let's explore these methods step by step! 🧐
Understanding Multiples
What are Multiples? 🤔
Multiples of a number are the results of multiplying that number by integers (whole numbers). For instance:
- The multiples of 15 are: 15, 30, 45, 60, 75, 90, ...
- The multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
Finding the Common Multiples 🔍
To find the LCM, we should look for the smallest common number in both lists. Let's list them out:
- Multiples of 15: 15, 30, 45, 60, 75, 90
- Multiples of 6: 6, 12, 18, 24, 30, 36
From these lists, we can see that the first common multiple is 30. Therefore, the LCM of 15 and 6 is 30.
Prime Factorization Method
What is Prime Factorization? 📊
Prime factorization involves breaking down a number into its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
Prime Factorization of 15 and 6
- 15 can be factored as ( 3 \times 5 ).
- 6 can be factored as ( 2 \times 3 ).
Combining the Factors 💡
To find the LCM using prime factorization, we need to take the highest power of each prime factor involved:
- From 15: ( 3^1 ) and ( 5^1 )
- From 6: ( 2^1 ) and ( 3^1 )
Now, taking the highest power from each factor:
| Prime Factor | Highest Power |
|---|---|
| 2 | ( 2^1 ) |
| 3 | ( 3^1 ) |
| 5 | ( 5^1 ) |
Now, multiply these together:
[ LCM = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 ]
Relationship between GCD and LCM
What is the Greatest Common Divisor (GCD)? 🧮
The GCD is the largest positive integer that divides both numbers without leaving a remainder. For our numbers:
- Factors of 15: 1, 3, 5, 15
- Factors of 6: 1, 2, 3, 6
The largest common factor is 3, so the GCD of 15 and 6 is 3.
LCM Formula 🔗
There is a relationship between GCD and LCM, which can be expressed as:
[ LCM(a, b) = \frac{|a \times b|}{GCD(a, b)} ]
Using this formula for our numbers:
[ LCM(15, 6) = \frac{15 \times 6}{GCD(15, 6)} = \frac{90}{3} = 30 ]
Conclusion: The LCM of 15 and 6 is 30 🎉
No matter which method you choose—listing multiples, prime factorization, or using the GCD—each confirms that the least common multiple of 15 and 6 is 30.
Summary Table
Here's a summary of the steps we took to find the LCM:
<table> <tr> <th>Method</th> <th>Process</th> <th>LCM Result</th> </tr> <tr> <td>Listing Multiples</td> <td>30 is the smallest common multiple</td> <td>30</td> </tr> <tr> <td>Prime Factorization</td> <td>(2^1, 3^1, 5^1) => 2 x 3 x 5</td> <td>30</td> </tr> <tr> <td>GCD Method</td> <td>(15 x 6) / GCD(15, 6)</td> <td>30</td> </tr> </table>
With this understanding of how to find the LCM of 15 and 6, you can apply similar techniques to any pair of numbers!