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Problem of the Day #12: LPofXaak Numbers March 31, 2011

Posted by Alex in : potd , trackback

A positive integer is LPofXaak if there exists a way to form two positive integers using its digits (for example, the pair $(8, 97)$ can be formed from $789$ and the pair $(01, 2)$ can be formed from $210$) such that the arithmetic mean of these two integers is equal to the geometric mean of the sum of the two integers and the positive difference of the two integers. Leading zeroes of numbers do not count as digits, and the numbers formed cannot contain leading zeroes. Let $Q$ be the sum of all LPofXaak numbers less than $1000$. Find the remainder when $Q$ is divided by $1000$.



1. Billy - March 31, 2011

wait i thought leading zeros couldn’t be used to form numbers. so how does a number like 350 work?

2. Alex - March 31, 2011

Oops. I need to make that clearer. Thanks Billy.
My original intention with that statement was: you can’t choose 16 and then make 10 and 6 by using the zero in front of the 16. I like your interpretation more, though.