Tuesday, November 12, 2019

elementary number theory - Divisibility and the gcd



Let a,b be integers with a|b (a divides b) and let a>0. Show that (a,b)=a. I know this is very basic, and that I'm complicating it unnecessarily, but for some reason I seem to be stuck...



Any help please?




Thanks.



Edit:
(a,b) means greatest common divisor of a and b


Answer



Well, recall the definition of (a,b): it's the integer that divides both a and b, and furthermore is the biggest integer that does so.



So how about asking:





  1. Does a divide a and b?

  2. Is a the biggest integer that does so?



Hopefully this makes it obvious.


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