Given that \( A=2^{3} \times 3 \) and \( B=2^{2} \times 3^{2} \) find the Lowest Common Multiple (LCM) of \( A \) and \( B \)

To find the Lowest Common Multiple (LCM) of \( A \) and \( B \), we start by breaking down \( A \) and \( B \) into their prime factorizations. Given: \[ A = 2^{3} \times 3^{1} \] \[ B = 2^{2} \times 3^{2} \] To find the LCM, we take the highest power of each prime factor from both factorizations. - For the prime \( 2 \): - The highest power in \( A \) is \( 2^{3} \). - The highest power in \( B \) is \( 2^{2} \). - Thus, we take \( 2^{3} \). - For the prime \( 3 \): - The highest power in \( A \) is \( 3^{1} \). - The highest power in \( B \) is \( 3^{2} \). - Thus, we take \( 3^{2} \). Now, we can compute the LCM: \[ \text{LCM}(A, B) = 2^{3} \times 3^{2} \] Calculating the final value: \[ 2^{3} = 8, \quad 3^{2} = 9 \] \[ \text{LCM}(A, B) = 8 \times 9 = 72 \] Therefore, the Lowest Common Multiple (LCM) of \( A \) and \( B \) is \( \boxed{72} \).