Two positive integers a and b can be written as ` a = x^(3) y^(3) and b = xy^(3) ` .x and y are prime numbers . Find LCM (a,b)

Answer

To find the LCM (Least Common Multiple) of the two positive integers \( a \) and \( b \) defined as:

- \( a = x^3 y^3 \)
- \( b = xy^3 \)

where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step 1: Write the prime factorization of \( a \) and \( b \)

For \( a = x^3 y^3 \):
- The prime factorization is \( x^3 \) and \( y^3 \).

For \( b = xy^3 \):
- The prime factorization is \( x^1 \) and \( y^3 \).

### Step 2: Identify the highest powers of each prime factor

To find the LCM, we take the highest power of each prime factor from both numbers:

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

- For the prime \( y \):
- In \( a \), the power is \( 3 \) (from \( y^3 \)).
- In \( b \), the power is \( 3 \) (from \( y^3 \)).
- The highest power is \( 3 \).

### Step 3: Write the LCM using the highest powers

Now, we can write the LCM as:
\[ \text{LCM}(a, b) = x^{\max(3, 1)} \cdot y^{\max(3, 3)} = x^3 \cdot y^3 \]

### Step 4: Finalize the LCM

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

### Summary of the Solution
The LCM of the integers \( a \) and \( b \) is \( x^3 y^3 \).

---