Problem:
If two positive integers a and b are written as a = x^3y^2 and b = xy^3, where x and y are prime numbers, then what is the highest common factor (HCF) of a and b? Explain in detail.

Solution:
To find the highest common factor (HCF) of two numbers a and b, we need to find the largest number that divides both a and b. In this case, we have two numbers a = x^3y^2 and b = xy^3, where x and y are prime numbers.

Step 1: Prime Factorization of a and b
To find the prime factorization of a and b, we need to express them as a product of prime numbers raised to their respective powers. Let's start with a:

a = x^3y^2

Here, a is expressed as the product of x raised to the power of 3 and y raised to the power of 2. Similarly, let's express b:

b = xy^3

Here, b is expressed as the product of x raised to the power of 1 and y raised to the power of 3.

Step 2: Finding the HCF
To find the HCF of a and b, we need to identify the common factors between the prime factorizations of a and b. In this case, x and y are the common factors.

Step 3: Determining the Powers of x and y
To find the HCF, we need to consider the lowest powers of the common factors. In this case, x and y are the common factors, and their respective powers are:

x: 1 (lowest power of x)
y: 2 (lowest power of y)

Step 4: Calculating the HCF
The HCF is calculated by multiplying the common factors raised to their lowest powers. In this case, the HCF is:

HCF(a, b) = x^1 * y^2
= xy^2

Therefore, the HCF of a and b is xy^2.

Summary:
Given two positive integers a = x^3y^2 and b = xy^3, where x and y are prime numbers, the HCF of a and b is xy^2. This is obtained by identifying the common factors (x and y) and taking the lowest powers of these factors. The HCF is then calculated by multiplying the common factors raised to their lowest powers.

X³y³