Lattice Points on Hypotenuse

Date: 10/01/2001 at 00:30:16
From: Alex
Subject: Number of Lattice points

Hello Dr. Math,

I just started eigth grade and am trying, without much success, to 
find the solution to the following problem:

What is the number of lattice points on the hypotenuse of a right 
angle triangle?

Can you please help?

Date: 10/04/2001 at 17:00:20
From: Doctor Peterson
Subject: Re: Number of Lattice points

Hi, Alex.

Suppose the legs have whole number lengths A and B; I'll assume the 
vertices are at (0,0), (A,0), and (A,B). (If it's a little different 
in the problem you were given, you can probably adjust my reasoning to 
fit.)

Suppose a lattice point (X,Y) is on the hypotenuse. Notice that, by 
similar triangles,

    A/B = X/Y

Since A, B, X, and Y are all whole numbers, you have two equivalent 
fractions. So think about how many equivalent fractions you can make; 
is there one for which X and Y are the smallest? What do you call 
that? If you find that one, how can you make the others using it?

What we've done is to change an unfamiliar problem into one that I 
think you will find much more familiar. That's how a lot of 
mathematical thinking works.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/