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## 'Procedure Solver' printed from http://nrich.maths.org/

Chanwc from Hong Kong, Matthew from QEB,
Alpha Lee from Yew Chung International School, Phil from Garforth
Community College and Andrei from Tudor Vianu National College all
correctly worked out that the algorithm produces the greatest
common factor of two whole numbers .

Aneesh correctly applied the algorithm for the cases X = 4 and Y =
1 and conjectured that the algorithm was to produce the output 1,
whereas Alpha started with this example:

initial x_i = 25 y_i = 5

1. x_1 = 25-5 = 20 y_1 = 5 since x_i> y_i

2. x_2 = 20-5 =15 y_2 = 5 since x_1> y_1

3. x_3 = 15-5 =10 y_3 = 5 since x_2> y_2

4. x_4 =10-5 =5 y_4 = 5 since x_2> y_2

5. x = output = 5 since x_4 = y_4 gcd(25,5) = 5 since 25 = 5*5 5 =
5*1

The algorithm (which was discovered by the
Greek mathematician Euclid over 2000 years ago) makes use of the
fact that the greatest common divisor of two numbers (a ,b) equals
the greatest common divisor of (a, b-a). Andre, Alpha and Phil all
noticed this fact, giving good, clear proofs. Phil's notation was
particularly clear :

Rename the two integers X and Y as AH and BH, meaning 'A multiples
of the HCF' and 'B multiples of the HCF'. The difference between
them, AH-BH, factorises to H(A-B). The difference between integers
X and Y must be an integer, as must the difference between A and B,
so the difference between X and Y must be a multiple of the HCF.
Phil then noted that As the algorithm cycles, the numbers X and Y
will always be replaced with lower multiples of the HCF of the
original X and Y. The algorithm continues to subtract until only
the HCF remains.

Matthew from QEB also solved the problem
and, interestingly, converted the procedure into Javascript, a
programming language most commonly used in webpages :

function getHCF(x,y) {

while (x != y) {

if (x > y) { x = x-y }

if (x < y) { y = y-x }

}

return x;

}

Matthew gave a clear and extended
description of his code, which you can read about here
.This is recommended for any budding
programmers!