Activity for future teachers and everyone else also

The Euclidean Algorithm that we explore in the lab has wonderful application to teaching and for those of you who will be teachers can be very valuable and useful.

The language of Mathematica in this unit is a bit complicated but do not get discouraged by this and here we will share some other approaches to the same idea.

You can use some of these approaches in your writings.

Imagine that you are a computer or a kind of machine that "eats" pairs of numbers (a,b) then a>=b and produces a new pair of numbers but immediately "eats" the new pair
unless one of the new numbers is 0.
The rules that the machine obeys are:
1. The machine stops "eating" when one of the numbers is 0.
2. If the machine tries to "eat" the pair (a,b) where a>=b>0 the new pair will be (b,c) where c is the remainder when dividing a by b.
Example:
(8,5) ---> (5,3) ----> (3,2) ---> (2,1) ---> (1,0). Note that 1=GCD(8,5) and it took five steps to reach the answer

This can be seen as a kind of game and one thing you can do is think how to design a lesson plan or an activity around it. For example dividing the numbers from 1 to 100 to families according to the number of steps.

Also how can we use a spread sheet to create this list.