What is the formula to find the least common multiple (LCM) of a set of numbers represented by $a_1,a_2,...,a_n$ in Latex format? Please demonstrate the application of this formula by finding the LCM of the following set of numbers: $\{8, 9, 12, 15\}$.
user
hello! i'm here to help you with your question. the formula to find the least common multiple (lcm) of a set of numbers represented by $a_1,a_2,...,a_n$ in latex format is:

$$\text{lcm}(a_1,a_2,...,a_n) = \prod_{i=1}^{n} a_i \cdot \frac{1}{\gcd(a_1,a_2,...,a_n)}$$

where $\gcd(a_1,a_2,...,a_n)$ represents the greatest common divisor of the set of numbers.

now, let's apply this formula to find the lcm of the set of numbers ${8, 9, 12, 15}$.

first, we find the greatest common divisor (gcd) of the set of numbers:

$$\gcd(8, 9, 12, 15) = \gcd(8, 12) \cdot \gcd(9, 15) = 4 \cdot 3 = 12$$

next, we multiply each number in the set by the gcd:

$$\prod_{i=1}^{4} a_i \cdot \frac{1}{\gcd(a_1,a_2,...,a_n)} = 8 \cdot \frac{1}{12} \cdot 9 \cdot \frac{1}{12} \cdot 12 \cdot \frac{1}{12} \cdot 15 \cdot \frac{1}{12} = \frac{8 \cdot 9 \cdot 12 \cdot 15}{12} = \frac{10080}{12} = 840$$

therefore, the least common multiple (lcm) of the set of numbers ${8, 9, 12, 15}$ is $840$.

i hope this helps you understand how to find the lcm of a set of numbers using the formula. if you have any more questions or need further clarification, please feel free to ask!
assistant
Hello! I'm here to help you with your question. The formula to find the least common multiple (LCM) of a set of numbers represented by $a_1,a_2,...,a_n$ in LaTeX format is:

$$\text{LCM}(a_1,a_2,...,a_n) = \prod_{i=1}^{n} a_i \cdot \frac{1}{\gcd(a_1,a_2,...,a_n)}$$

where $\gcd(a_1,a_2,...,a_n)$ represents the greatest common divisor of the set of numbers.

Now, let's apply this formula to find the LCM of the set of numbers ${8, 9, 12, 15}$.

First, we find the greatest common divisor (GCD) of the set of numbers:

$$\gcd(8, 9, 12, 15) = \gcd(8, 12) \cdot \gcd(9, 15) = 4 \cdot 3 = 12$$

Next, we multiply each number in the set by the GCD:

$$\prod_{i=1}^{4} a_i \cdot \frac{1}{\gcd(a_1,a_2,...,a_n)} = 8 \cdot \frac{1}{12} \cdot 9 \cdot \frac{1}{12} \cdot 12 \cdot \frac{1}{12} \cdot 15 \cdot \frac{1}{12} = \frac{8 \cdot 9 \cdot 12 \cdot 15}{12} = \frac{10080}{12} = 840$$

Therefore, the least common multiple (LCM) of the set of numbers ${8, 9, 12, 15}$ is $840$.

I hope this helps you understand how to find the LCM of a set of numbers using the formula. If you have any more questions or need further clarification, please feel free to ask!
assistant