1. **State the problem:** Find the lowest common multiple (LCM) of the numbers $$A = 2^3 \times 3 \times 5 \times 7^2$$ $$B = 2 \times 3^2 \times 7$$ $$C = 3 \times 5^2 \times 11$$ 2. **Recall the formula for LCM:** The LCM of several numbers is found by taking the highest power of each prime factor appearing in any of the numbers. 3. **Identify prime factors and their powers:** - For 2: max power is $\max(3,1,0) = 3$ - For 3: max power is $\max(1,2,1) = 2$ - For 5: max power is $\max(1,0,2) = 2$ - For 7: max power is $\max(2,1,0) = 2$ - For 11: max power is $\max(0,0,1) = 1$ 4. **Write the LCM as product of prime powers:** $$\text{LCM} = 2^3 \times 3^2 \times 5^2 \times 7^2 \times 11$$ 5. **Final answer:** The lowest common multiple of $A$, $B$, and $C$ is $$\boxed{2^3 \times 3^2 \times 5^2 \times 7^2 \times 11}$$