UpStudy Free Solution:

To find the Least Common Multiple (LCM) of 6, 10, and 15, we can use the list of multiples method you've started or the prime factorization method. Let's use the list of multiples method first.

List of Multiples Method

1. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...

2. Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, ...

3. Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, ...

From the lists, we can see that the common multiples of 6, 10, and 15 are 30 and 60, etc. The smallest common multiple is 30.

So, the LCM of 6, 10, and 15 is 30.

Prime Factorization Method

Alternatively, let's use the prime factorization method to verify:

1. Prime factorization of 6: \(6 = 2 \times 3\)

2. Prime factorization of 10: \(10 = 2 \times 5\)

3. Prime factorization of 15: \(15 = 3 \times 5\)

To find the LCM, we take the highest power of each prime that appears in the factorizations:

- The highest power of 2: \(2^ 1\)

- The highest power of 3: \(3^ 1\)

- The highest power of 5: \(5^ 1\)

Now, multiply these together:

\[LCM = 2^ 1 \times 3^ 1 \times 5^ 1 = 2 \times 3 \times 5 = 30\]

So, the LCM of 6, 10, and 15 is 30.

Supplemental Knowledge

Understanding the concept of the Least Common Multiple (LCM) is essential in various mathematical applications, such as solving problems involving fractions, algebraic expressions, and more. Here are additional details on methods to find the LCM:

1. List of Multiples Method:

- This method involves listing the multiples of each number until a common multiple is found.

- It is straightforward but can be time-consuming for larger numbers.

2. Prime Factorization Method:

- This method involves breaking down each number into its prime factors.

- The LCM is found by taking the highest power of each prime factor that appears in any of the factorizations.

- This method is efficient and works well for both small and large numbers.

3. Division Method (also known as Ladder Method):

- This method involves dividing the given numbers by their common prime factors until all resulting quotients are 1.

- The LCM is then obtained by multiplying all the divisors used.

Example Using Division Method:

To find the LCM of 6, 10, and 15 using the division method:

1. Write down the numbers in a row:

\[\begin{array} { c| ccc} & 6 & 10 & 15 \\ \hline 2 & 3 & 5 & 15 \\ \hline 3 & 1 & 5 & 5 \\ \hline 5 & 1 & 1 & 1 \\ \end{array} \]

2. Divide by common prime factors until all quotients are reduced to one.

3. Multiply all divisors used:

\[LCM = 2 \times 3 \times 5 = 30\]

So, using this method also confirms that the LCM of \(6\), \(10\), and \(15\) is \(30\).

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