Two positive integer a and b are written as `a = x^(3)y^(2), b = xy^(3)` , the LCM of (a,b) is

Options

A. `x^(3)y^(3)`
B. `xy^(2)`
C. `x^(2)y^(2)`
D. `xy^(3)`

Answer

To find the LCM (Lowest Common Multiple) of the two positive integers \( a \) and \( b \) given as \( a = x^3 y^2 \) and \( b = x y^3 \), we will follow these steps:

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

- For \( a = x^3 y^2 \):
- The prime factors are \( x \) raised to the power of 3 and \( y \) raised to the power of 2.

- For \( b = x y^3 \):
- The prime factors are \( x \) raised to the power of 1 and \( y \) raised to the power of 3.

### Step 2: Determine the LCM

The LCM is found by taking the highest power of each prime factor present in both numbers.

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

- For the prime factor \( y \):
- The highest power is \( \max(2, 3) = 3 \).

### Step 3: Write the LCM

Now, we can write the LCM using the highest powers determined:

\[ \text{LCM}(a, b) = x^3 y^3 \]

### Conclusion

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

\[ \text{LCM}(a, b) = x^3 y^3 \]

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