If positive integers a and bare written as `a= xy^(2) and b= x^2y,` wherex, y are prime numbers, then find LCM (a, b)

Answer

To find the LCM of the positive integers \( a \) and \( b \) given as \( a = xy^2 \) and \( b = x^2y \), where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step 1: Prime Factorization
First, we express \( a \) and \( b \) in terms of their prime factors.

- For \( a = xy^2 \):
- This can be written as \( a = x \cdot y \cdot y \) (which means \( x^1 \) and \( y^2 \)).

- For \( b = x^2y \):
- This can be expressed as \( b = x \cdot x \cdot y \) (which means \( x^2 \) and \( y^1 \)).

### Step 2: Identify Maximum Powers of Each Prime Factor
Next, we identify the maximum power of each prime factor that appears in the factorizations of \( a \) and \( b \).

- For the prime \( x \):
- In \( a \), the power of \( x \) is \( 1 \).
- In \( b \), the power of \( x \) is \( 2 \).
- The maximum power of \( x \) is \( 2 \).

- For the prime \( y \):
- In \( a \), the power of \( y \) is \( 2 \).
- In \( b \), the power of \( y \) is \( 1 \).
- The maximum power of \( y \) is \( 2 \).

### Step 3: Calculate the LCM
The LCM is found by taking the product of each prime factor raised to its maximum power:

\[ \text{LCM}(a, b) = x^{\text{max}(1, 2)} \cdot y^{\text{max}(2, 1)} = x^2 \cdot y^2 \]

Thus, the LCM of \( a \) and \( b \) is:

\[ \text{LCM}(a, b) = x^2y^2 \]

### Summary of the Solution
The LCM of the integers \( a \) and \( b \) is \( x^2y^2 \).

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