Assignment on Use properties of Farey's sequence to prove the GCD theorem

The proper arbitrary fraction of p/q is used to find the closest fraction in the binary search that reduce the search range of ranks. The properties of Farey’s sequences are used to prove the GCD theorem here. The first step is to measure the range of the sequence that can be obtained by different ways. Suppose the search key is p/q and range become . Consider two properties of Farey sequence for gcd theorem.

The fractions are now in the ascending order of  and the binary search provides the close values of  to get the fraction that is close to p/q in the function.

Using the property 3 for the maximum rank in the function it gives

Considering the corresponding binary search for the key fraction. The plots define the gcd theorem with fraction difference and key values are under the rank defined above that is  as a key fraction p/q. from the explicit evaluation of function and theorem it gives values p and q. consider that p and q are both coprime and c  = 1 the rule become as c =  gcd (p, q)