If two positive integers a and b are written as a = x3y2 and b = xy3; where x, y areprime numbers, then HCF (a,b) is:

To find the highest common factor (HCF) of two positive integers a and b, which are written as a = x^3y^2 and b = xy^3, where x and y are prime numbers, we can follow these steps:

Step 1: Prime factorize both a and b.

• For a = x^3y^2, the prime factors are x and y, with exponents 3 and 2 respectively.
• For b = xy^3, the prime factors are x and y, with exponents 1 and 3 respectively.

Step 2: Identify the common prime factors and their lowest exponents.

• The common prime factors are x and y.
• The lowest exponent for x is 1, as it appears with exponent 1 in b.
• The lowest exponent for y is 2, as it appears with exponent 2 in a.

Step 3: Calculate the HCF.

• The HCF is obtained by multiplying the common prime factors with their lowest exponents.
• Therefore, the HCF of a and b is x^1y^2, which simplifies to xy^2.

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