Two positive integers expressed as `p=ab^2` and `q=a^2b`, where a and b are prime numbers. Then the LCM of p and q is:

Options

A. `b^2`
B. ab
C. `a^3b^3`
D. `a^2b^2`

Answer

To find the LCM of the two positive integers \( p = ab^2 \) and \( q = a^2b \), where \( a \) and \( b \) are prime numbers, we can follow these steps:

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

### Step 2: Identify the prime factorization of \( p \) and \( q \)
- For \( p = ab^2 \), the prime factorization is:
- \( a^1 \) (the power of \( a \) is 1)
- \( b^2 \) (the power of \( b \) is 2)

- For \( q = a^2b \), the prime factorization is:
- \( a^2 \) (the power of \( a \) is 2)
- \( b^1 \) (the power of \( b \) is 1)

### Step 3: Determine the highest powers of each prime factor
Now, we need to find the highest power of each prime factor present in both \( p \) and \( q \):
- For \( a \):
- The highest power in \( p \) is \( 1 \) (from \( p = a^1b^2 \))
- The highest power in \( q \) is \( 2 \) (from \( q = a^2b^1 \))
- Therefore, the highest power of \( a \) is \( 2 \).

- For \( b \):
- The highest power in \( p \) is \( 2 \) (from \( p = a^1b^2 \))
- The highest power in \( q \) is \( 1 \) (from \( q = a^2b^1 \))
- Therefore, the highest power of \( b \) is \( 2 \).

### Step 4: Write the LCM using the highest powers
The LCM is calculated by taking the highest powers of all prime factors:
\[ \text{LCM}(p, q) = a^{\text{highest power of } a} \times b^{\text{highest power of } b} = a^2 \times b^2 \]

### Final Answer
Thus, the LCM of \( p \) and \( q \) is:
\[ \text{LCM}(p, q) = a^2b^2 \]

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