lcm of 8, 12, and 15

Understand the Problem

The question is asking to find the least common multiple (LCM) of the numbers 8, 12, and 15. To solve it, we will identify the multiples of each number and determine the smallest multiple that is common to all three.

Answer

$120$

Steps to Solve

Find the prime factorization of each number

Start by breaking down each number into its prime factors.
- For 8: $$ 8 = 2^3 $$
- For 12: $$ 12 = 2^2 \times 3^1 $$
- For 15: $$ 15 = 3^1 \times 5^1 $$
Identify the highest power of each prime factor

List all the prime factors found from the factorization and take the highest power for each.
- From 8: $2^3$
- From 12: $2^2$ and $3^1$
- From 15: $3^1$ and $5^1$
The highest powers are:
- $2^3$
- $3^1$
- $5^1$
Calculate the LCM using the highest powers

Multiply the highest powers of all prime factors to find the LCM.

So, we calculate: $$ LCM = 2^3 \times 3^1 \times 5^1 $$

We can compute this step by step:
- First, calculate $2^3 = 8$
- Then, $3^1 = 3$
- Finally, $5^1 = 5$
So, we multiply them together: $$ LCM = 8 \times 3 \times 5 $$
Final multiplication to find the least common multiple

Now we calculate: $$ 8 \times 3 = 24 $$ $$ 24 \times 5 = 120 $$

Therefore, the least common multiple of 8, 12, and 15 is 120.

The least common multiple (LCM) of 8, 12, and 15 is $120$.

More Information

The least common multiple is a useful concept in various mathematical applications, including finding common denominators in fractions and solving problems related to periods and cycles. The LCM can help in determining the smallest number that is a multiple of given numbers, especially in scheduling and planning.

Tips

Common mistakes include:

Forgetting to use the highest power of all prime factors—always check each factor!
Mixing up multiplication of factors—take your time to calculate step by step to avoid errors.

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