If two positive integers a and b are written as $\mathrm{a}={\mathrm{x}}^3{\mathrm{y}}^2$ and $\mathrm{b}={\mathrm{xy}}^2$; x and y are prime numbers, then HCF (a, b) is:

Two integers $\mathrm{a}={\mathrm{x}}^3{\mathrm{y}}^2$ and $\mathrm{b}={\mathrm{xy}}^2$ are given and we need to find their HCF.

The given numbers can be written as the product of their prime factors as follows:

$\begin{array}{l}{\mathrm{x}}^3{\mathrm{y}}^2=\mathrm{x}\times \mathrm{x}\times \mathrm{x}\times \mathrm{y}\times \mathrm{y}\\ {\mathrm{xy}}^2=\mathrm{x}\times \mathrm{y}\times \mathrm{y}\end{array}$

From the above prime factorization, it can be observed that the common prime factors are x, y and y.

Since the HCF is the product of the common prime factors, HCF of x³y² and xy² is given by

$\begin{array}{l}\mathrm{HCF}=\mathrm{x}\times \mathrm{y}\times \mathrm{y}\\ \Rightarrow \mathrm{HCF}={\mathrm{xy}}^2\end{array}$

Hence, the HCF of x³y³ and xy² is xy².