The LCM and the HCF of \( 5 a^{2} b \) and \( 3 a b^{2} \) respectively are:

To find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of the expressions \( 5 a^{2} b \) and \( 3 a b^{2} \), we will follow these steps: 1. **Identify the prime factorization of each term.** 2. **Determine the HCF.** 3. **Determine the LCM.** ### Step 1: Prime Factorization - For \( 5 a^{2} b \): - The prime factors are \( 5 \), \( a^{2} \), and \( b \). - For \( 3 a b^{2} \): - The prime factors are \( 3 \), \( a \), and \( b^{2} \). ### Step 2: Finding the HCF The HCF is found by taking the lowest power of each common factor. - For the numerical coefficients: - The common factor is \( 1 \) (since \( 5 \) and \( 3 \) have no common factors). - For \( a \): - The lowest power is \( a^{1} \). - For \( b \): - The lowest power is \( b^{1} \). Thus, the HCF is: \[ \text{HCF} = 1 \cdot a^{1} \cdot b^{1} = a b \] ### Step 3: Finding the LCM The LCM is found by taking the highest power of each factor. - For the numerical coefficients: - The LCM of \( 5 \) and \( 3 \) is \( 15 \). - For \( a \): - The highest power is \( a^{2} \). - For \( b \): - The highest power is \( b^{2} \). Thus, the LCM is: \[ \text{LCM} = 15 \cdot a^{2} \cdot b^{2} = 15 a^{2} b^{2} \] ### Final Results - The HCF of \( 5 a^{2} b \) and \( 3 a b^{2} \) is \( a b \). - The LCM of \( 5 a^{2} b \) and \( 3 a b^{2} \) is \( 15 a^{2} b^{2} \).