Question

Find the least common multiple of 24x and 30y

Answer 1

To find the least common multiple (LCM) of two expressions involving variables, we first need to determine the prime factorization of the numerical coefficients and then account for the variable parts.

1. **Factorize the coefficients:**
- The prime factorization of \( 24 \) is:
\[
24 = 2^3 \times 3^1
\]
- The prime factorization of \( 30 \) is:
\[
30 = 2^1 \times 3^1 \times 5^1
\]

2. **Determine the LCM of the coefficients:**
- For the LCM, we take the highest power of each prime factor:
- For \( 2 \): The highest power is \( 2^3 \).
- For \( 3 \): The highest power is \( 3^1 \).
- For \( 5 \): The highest power is \( 5^1 \).

Thus, the LCM of the coefficients \( 24 \) and \( 30 \) is:
\[
LCM(24, 30) = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5
\]

Calculating this gives:
\[
8 \times 3 = 24
\]
\[
24 \times 5 = 120
\]

3. **Combine the variable parts:**
- The LCM must also include the variable parts. Since we are finding the LCM of \( 24x \) and \( 30y \), we simply take both variables \( x \) and \( y \):
- We take \( x^1 \) and \( y^1 \) as both are raised to the first power.

4. **Combine everything together:**
The LCM of \( 24x \) and \( 30y \) is:
\[
LCM(24x, 30y) = 120xy
\]

Thus, the least common multiple of \( 24x \) and \( 30y \) is \( \boxed{120xy} \).