Challange Problem.

Prove that if and only if and share exactly the same prime divisors.

Moderator approved CB

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- Sep 30th 2010, 11:00 PMmeleseEuler's phi-function.
**Challange Problem.**

Prove that if and only if and share exactly the same prime divisors.

Moderator approved CB - Oct 1st 2010, 12:58 AMsimplependulum
Let and ( prime factorization of and respectively . )

We know , it is obvious that if and share the same prime divisors , the product should equal as we can find a bijection linking and .

Suppose for , they don't share all the same prime divisors , let , where is the greatest common divisor of them . Now and are relatively prime and not equal to ( if so , the other must also be , then , a contradiction . )

Let and be the prime factorization of and respectively .

Consider ,

since are distinct , we must have and . and are less than but in this case they are positive integers , this leads to a contradiction . - Oct 1st 2010, 05:37 AMmelese
- Oct 4th 2010, 12:08 PMBruno J.
- Oct 5th 2010, 08:29 AMBruno J.
Alright. Let be such that . As simplependulum remarked, we can write this equation as

,

where denote, respectively, the sets of prime divisors of . Now we can divide both members by

,

after which we are left with

,

where and (so that ). Now suppose and take any . Since , we must have , i.e. for some . But if you take the greatest , this is clearly impossible. Hence , and similarily , hence . - Oct 5th 2010, 10:37 AMBruno J.
What I wrote is perhaps what S.P. had in mind.