Wednesday, 25 December 2013

elementary number theory - if a,b are both integers and coprime, prove that the gcd(a2b3,a+b)=1




I'm trying to solve this problem. I should be able to do it using simple divisibility properties but I don't know how.




Let a and b be integers such that they are coprime. Prove that gcd




For instance... I thought that the gcd divides both a^2b^3 and a+b so it must divide a sum of them. I've tried going this way but it's not clear to me where it should lead me. Any hint will be welcomed. Thanks.


Answer



Suppose that p is a prime number such that p|a^2b^3 then p|a or p|b. Let's say p|a. If p|(a+b) then we should have p|b what is impossible because a,b are coprimes.



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