The following are prime factors of certain numbers. Who is their LCM?

To find the Least Common Multiple (LCM) of numbers given in their prime factorized form, we take the highest power of each prime factor present in any of the numbers.

The given numbers are:

1. $2^2 \times 3^2$
2. $2^3 \times 5$
3. $2^4 \times 3^2$

Step 1: Identify all unique prime factors present in the numbers. The unique prime factors are 2, 3, and 5.

Step 2: Determine the highest power for each unique prime factor across all numbers.

• For prime factor 2: The powers are $2^2$, $2^3$, and $2^4$. The highest power is $2^4$.
• For prime factor 3: The powers are $3^2$, $3^0$ (from the second number), and $3^2$. The highest power is $3^2$.
• For prime factor 5: The powers are $5^0$ (from the first number), $5^1$, and $5^0$ (from the third number). The highest power is $5^1$.

Step 3: Multiply these highest powers together to find the LCM. $\text{LCM} = 2^4 \times 3^2 \times 5^1$

Step 4: Calculate the value. $\text{LCM} = 16 \times 9 \times 5$ $\text{LCM} = 144 \times 5$ $\text{LCM} = \boxed{720}$