# coprime problem

• Aug 2nd 2012, 07:56 AM
lokeshkumargupta
coprime problem
pls solve this problem for me as m not also to solve it...

question: suppose a and b are relatively prime. prove that ab and a+b are relatively prime...
• Aug 3rd 2012, 01:39 AM
kraj8995
Re: coprime problem
suppose a=3
b=5
ab=5*3=15
a+b=8

so a+b is not a relatively prime.
• Aug 3rd 2012, 01:44 AM
emakarov
Re: coprime problem
Quote:

Originally Posted by kraj8995
so a+b is not a relatively prime.

You cannot say "relatively prime" about one number.
• Aug 3rd 2012, 03:55 AM
lokeshkumargupta
Re: coprime problem
two numbers a,b are said to be coprime if gcd(a,b)=1
• Aug 3rd 2012, 03:57 AM
Deveno
Re: coprime problem
one could suppose that a counter-example existed. if so then since d > 1 divides ab and a+b, there is some prime p that divides d. but p, being prime, must divide a or b.

suppose it were a, that is a = kp. since p also divides a+b (since d does), a+b = rp. hence b = a+b - a = rp - kp = (r-k)p, so p divides b.

or, suppose it were b, so that b = mp. then a = a+b - b = rp - mp = (r-m)p, and so a is likewise divisible by p.

either way, assuming that gcd(ab,a+b) > 1 leads to a prime p dividing a and b. but gcd(a,b) = 1, so how can this be possible?