If two positive integers a and b are written as `a=x^4y^2 and b=x^(3)y`, where x, y are prime numbers, then find HCF (a, b).

Answer

To find the HCF (Highest Common Factor) of the two positive integers \( a \) and \( b \) defined as \( a = x^4y^2 \) and \( b = x^3y \), where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step 1: Prime Factorization
We start by expressing \( a \) and \( b \) in terms of their prime factors.

- For \( a \):
\[ a = x^4 y^2 = x \cdot x \cdot x \cdot x \cdot y \cdot y \]

- For \( b \):
\[ b = x^3 y = x \cdot x \cdot x \cdot y \]

### Step 2: Identify Common Factors
Next, we identify the common prime factors in both \( a \) and \( b \).

- For \( x \):
- In \( a \), we have \( x^4 \) (which means there are 4 factors of \( x \)).
- In \( b \), we have \( x^3 \) (which means there are 3 factors of \( x \)).
- The common factors of \( x \) will be the minimum of the two, which is \( x^3 \).

- For \( y \):
- In \( a \), we have \( y^2 \) (which means there are 2 factors of \( y \)).
- In \( b \), we have \( y^1 \) (which means there is 1 factor of \( y \)).
- The common factors of \( y \) will be the minimum of the two, which is \( y^1 \).

### Step 3: Multiply Common Factors
Now, we multiply the common factors to find the HCF.

\[ \text{HCF}(a, b) = x^3 \cdot y^1 = x^3 y \]

Thus, the HCF of \( a \) and \( b \) is \( x^3 y \).

### Final Answer
\[ \text{HCF}(a, b) = x^3 y \]

---