If two positive integers a and b can be expressed as `a=x^(2) y^(5) and b=x^(3) y^(2), where x,y` are prime numbers, then find LCM of `a and b`.

Answer

To find the LCM of the two positive integers \( a \) and \( b \) expressed as \( a = x^2 y^5 \) and \( b = x^3 y^2 \), where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step 1: Identify the expressions for \( a \) and \( b \)
We have:
- \( a = x^2 y^5 \)
- \( b = x^3 y^2 \)

### Step 2: Determine the highest power of each prime factor
To find the LCM, we need to take the highest power of each prime factor from both \( a \) and \( b \).

1. **For \( x \)**:
- In \( a \), the power of \( x \) is \( 2 \).
- In \( b \), the power of \( x \) is \( 3 \).
- The highest power of \( x \) is \( 3 \) (from \( b \)).

2. **For \( y \)**:
- In \( a \), the power of \( y \) is \( 5 \).
- In \( b \), the power of \( y \) is \( 2 \).
- The highest power of \( y \) is \( 5 \) (from \( a \)).

### Step 3: Write the LCM using the highest powers
Now that we have the highest powers:
- For \( x \), the highest power is \( x^3 \).
- For \( y \), the highest power is \( y^5 \).

Thus, the LCM of \( a \) and \( b \) is:
\[ \text{LCM}(a, b) = x^3 y^5 \]

### Final Answer
The LCM of \( a \) and \( b \) is \( x^3 y^5 \).

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