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

Options

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

Answer

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

### Step 1: Express \( p \) and \( q \) in terms of their prime factors
- Given:
- \( p = ab^2 \)
- \( q = a^3b \)

### Step 2: Identify the prime factors and their powers
- For \( p \):
- The prime factor \( a \) has a power of \( 1 \) (since it appears once).
- The prime factor \( b \) has a power of \( 2 \) (since it appears as \( b^2 \)).

So, we can write:
\[ p = a^1 b^2 \]

- For \( q \):
- The prime factor \( a \) has a power of \( 3 \) (since it appears as \( a^3 \)).
- The prime factor \( b \) has a power of \( 1 \) (since it appears once).

So, we can write:
\[ q = a^3 b^1 \]

### Step 3: Determine the LCM using the highest powers of the prime factors
- The LCM is found by taking the highest power of each prime factor from both \( p \) and \( q \).

- For the prime factor \( a \):
- The highest power is \( \max(1, 3) = 3 \).

- For the prime factor \( b \):
- The highest power is \( \max(2, 1) = 2 \).

### Step 4: Write the LCM
- Therefore, the LCM of \( p \) and \( q \) is:
\[ \text{LCM}(p, q) = a^3 b^2 \]

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

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