A Solution to the "Ten-Minute" Problem

It should be easy to observe that z must be odd. We are looking for positive integer solution to



By manually doing division algorithm once, you can show that . Simultaneously, you can find a particular integer solution to , which is




Now, set Then we have




as general solution. To keep x and y both positive, we may have a "sense" that when is small, we can adjust K to keep x small but positive, and hence "allow room" for y to be positive.

Take , we have . However, . The remainder 1651 is too large to be x.

Now observe that when z is increased by 2, a drops by 4609. Notice . That means when we increase z by 2, drops by 537.

Increase z from 1 to 7, drops from 1651 to . Setting , we get . A possible solution is given by