# If x and y are positive integers such that the greatest common factor of x^2y^2 and xy^3 is 45, then which of the following could y equal?

Jun 12, 2016

No answers were provided from which to select;
the only possible answers are $y = 1$ or $y = 3$

#### Explanation:

${x}^{2} {y}^{2}$ and $x {y}^{3}$ have a common factor of $x {y}^{2}$

If $x$ and $y$ are positive integers any common factor of ${x}^{2} {y}^{2}$ and $x {y}^{3}$ must itself have a factor of ${y}^{2}$

If $45$ is the greatest common factor (or any factor) of ${x}^{2} {y}^{2}$ and $x {y}^{3}$
then $45$ must contain ${y}^{2}$ as a factor.

Restricting ourselves to positive factors:
$\textcolor{w h i t e}{\text{XXX}} 45 = {1}^{2} \times {3}^{2} \times 5$

So $y$ must be either $1$ or $3$