Thursday, 1 December 2016

number theory - How to compute these GCD's?



Please suggest me how to compute the GCD of theese really big numbers :





  1. GCD of 21205475643971 and 23569466819401

  2. GCD of 2n1 and n! where n=319



Thanks to Bill Dubuque's answer I understood that the first problem could be solved by the property that gcd(f(m),f(n))=f(gcd(m,n)) if f(n)f(nm)(mod f(m)),   f(0) = 0.



Any hints for the second one?


Answer




HINT  (2a1,2b1) = 2(a,b)1  where  (a,b):=gcd(a,b).  For proofs see here, or here - which has a polynomial analog.


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