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If two positive integers P and q can be expressed as `p=a^2b^3 and q=a^4 b,ab` being prime numbers then LCM (p,q) is……

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To find the LCM of the two positive integers \( P \) and \( Q \) given as \( P = a^2 b^3 \) and \( Q = a^4 b \), we can follow these steps: ### Step 1: Identify the prime factorization of \( P \) and \( Q \) - For \( P = a^2 b^3 \): - The prime factor \( a \) has an exponent of 2. - The prime factor \( b \) has an exponent of 3. - For \( Q = a^4 b^1 \): - The prime factor \( a \) has an exponent of 4. - The prime factor \( b \) has an exponent of 1. ### Step 2: Determine the LCM using the highest powers of each prime factor - For the prime factor \( a \): - The highest power between \( a^2 \) (from \( P \)) and \( a^4 \) (from \( Q \)) is \( a^4 \). - For the prime factor \( b \): - The highest power between \( b^3 \) (from \( P \)) and \( b^1 \) (from \( Q \)) is \( b^3 \). ### Step 3: Combine the highest powers to find the LCM - The LCM of \( P \) and \( Q \) is given by: \[ \text{LCM}(P, Q) = a^4 b^3 \] ### Final Answer: \[ \text{LCM}(P, Q) = a^4 b^3 \] ---
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