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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