## Product of X and Y

$X \textup{ and }Y$ are whole numbers and neither of which are divisible by 10. If the product of $X \textup{ and }Y$ equals 20000, find the absolute value of the difference of $X \textup{ and }Y$.
Source: mathcontest.olemiss.edu 12/1/2008

SOLUTION
This problem is similar to the problem titled “100000” dated 1/12/2009.
$20000=2\times 10\times 10\times 10\times 10$
$=2\times 2\times 5\times 2\times 5\times 2\times 5\times 2\times 5$
$=2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\times 5$
$= X\times Y$

Since we want $X \textup { and }Y$ not divisible by 10, we should keep the 2’s and 5’s separate when we form the two factors $X\textup{ and }Y$ of 20000. Thus,
$X=2\times 2\times 2\times 2\times 2$
$=32$
$Y=5\times 5\times 5\times 5$
$=625$

$\left |X-Y\right |=\left |32-625\right |$
$=\left |-593\right |$
$=593$.