If two positive integers p and q can be expressed as `p=ab^2 and q=a^3b` where a and b are prime numbers, then the LCM (p,q) is

Options

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

Answer

To find the LCM of the two positive integers \( p \) and \( q \) given as \( p = ab^2 \) and \( q = a^3b \), where \( a \) and \( b \) are prime numbers, we can follow these steps:

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

### Step 2: Factor \( p \) and \( q \)
Let's express \( p \) and \( q \) in terms of their prime factorization:
- For \( p = ab^2 \):
- The prime factorization is \( a^1 \cdot b^2 \).

- For \( q = a^3b \):
- The prime factorization is \( a^3 \cdot b^1 \).

### Step 3: Identify the highest powers of each prime factor
To find the LCM, we need to take the highest power of each prime factor that appears in the factorizations of \( p \) and \( q \):
- For the prime \( a \):
- The highest power is \( a^3 \) (from \( q \)).

- For the prime \( b \):
- The highest power is \( b^2 \) (from \( p \)).

### Step 4: Write the LCM
Now, we can write the LCM of \( p \) and \( q \) using the highest powers identified:
\[ \text{LCM}(p, q) = a^3 \cdot b^2 \]

### Conclusion
Thus, the LCM of \( p \) and \( q \) is \( a^3b^2 \).

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