Two positive integers p and q are expressible as `p=a^(3)b` and `q=ab^(2)`. Find the HCF (p,q) and LCM(p,q).

Answer

To solve the problem of finding the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the integers \( p \) and \( q \), given that \( p = a^3 b \) and \( q = a b^2 \), we can follow these steps:

### Step 1: Write down the expressions for \( p \) and \( q \)
Given:
- \( p = a^3 b \)
- \( q = a b^2 \)

### Step 2: Factorize \( p \) and \( q \)
We can express \( p \) and \( q \) in terms of their prime factors:
- \( p = a^3 \cdot b^1 \)
- \( q = a^1 \cdot b^2 \)

### Step 3: Find the HCF (Highest Common Factor)
To find the HCF, we take the lowest power of each common factor:
- For \( a \): The powers are \( 3 \) (from \( p \)) and \( 1 \) (from \( q \)). The minimum is \( 1 \).
- For \( b \): The powers are \( 1 \) (from \( p \)) and \( 2 \) (from \( q \)). The minimum is \( 1 \).

Thus, the HCF is:
\[ \text{HCF}(p, q) = a^{\min(3, 1)} \cdot b^{\min(1, 2)} = a^1 \cdot b^1 = ab \]

### Step 4: Find the LCM (Lowest Common Multiple)
To find the LCM, we take the highest power of each factor:
- For \( a \): The powers are \( 3 \) (from \( p \)) and \( 1 \) (from \( q \)). The maximum is \( 3 \).
- For \( b \): The powers are \( 1 \) (from \( p \)) and \( 2 \) (from \( q \)). The maximum is \( 2 \).

Thus, the LCM is:
\[ \text{LCM}(p, q) = a^{\max(3, 1)} \cdot b^{\max(1, 2)} = a^3 \cdot b^2 \]

### Final Answers
- HCF of \( p \) and \( q \) is \( ab \).
- LCM of \( p \) and \( q \) is \( a^3 b^2 \).