 QUESTION

# If two positive integers a and b are written as a = ${x^2}{y^2}$ and b = $x{y^2}$, where $x$ and $y$ are the prime numbers then the HCF (a, b) is

Hint: Prime factors of a and b are given. So, let us only use the definition of HCF which states that HCF of two numbers is the highest common factor that these numbers can have.

As we know that to find HCF of two numbers first we have to split both numbers into products of prime factors of that number.
And then find the product of factors that are common to both the numbers because HCF of two numbers is the highest common factor both numbers can have.
So, now we are given that $x$ and $y$ are prime numbers.
And, a = ${x^2}{y^2}$
So, when we write a as the product of its factors then it becomes,
a $= x \times x \times y \times y$ (1)
And, b = $x{y^2}$
So, b as the product of its prime factors will be.
b = $x \times y \times y$ (2)
Now from equation 1 and 2. We can see that the maximum common factor that a and b can have will be $x \times y \times y$.
Hence, the HCF of a and b will be $x{y^2}$.

Note: Whenever we come up with this type of problem then we should understand that it does not depend that factors are any given integer of any variable. HCF will be the highest common factor of both numbers either it will be variable or any integer. So, we should write the numbers as the product of prime factors and then take common factors out of them. And the product of common factors will be the required HCF of the given numbers.