## 100000

The variables $a \textup{ and }b$ represent positive integers, neither of which are divisible by 10 yet the product of $a\textup{ and }b$ is 100000.  Find the sum of $a\textup{ and }b$.
Source: mathcontest.olemiss.edu 1/12/2009

SOLUTION
This problem is similar to the problem titled “Product of X and Y” dated 12/1/2008.
$100000=10\times 10\times 10\times 10\times 10$
$=2\times 5\times 2\times 5\times 2\times 5\times 2\times 5\times 2\times 5$
$=2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\times 5\times 5$
$=a\times b$

Since we want $a \textup{ and }b$ not divisible by 10, we should keep the 2’s and the 5’s separate when we form the two factors $a \textup{ and }b$ of 100000. Thus,
$a=2\times 2\times 2\times 2\times 2$
$=32$

$b=5\times 5\times 5\times 5\times 5$
$=3125$

The sum $a+b=32+3125=3157$.