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Topic: ------ ------ ----- coprime integers
Replies: 2   Last Post: Sep 27, 2011 7:35 PM

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Bill Dubuque

Posts: 1,739
Registered: 12/6/04
Re: ------ ------ ----- coprime integers
Posted: Sep 27, 2011 7:35 PM
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Deep <deepkdeb@yahoo.com> wrote:
>
> Let a, b be two coprime integers such that 2|b
> Statement: a, b, a+b, a - b are all coprime integers.
>
> Justification of the statement:
>
> There exista one prime p such that p|a
> Since p|a and p doesnot divide b then p cannot divide (a + b).
> By similar argument p does not divide(a - b).
> Consequently, a, b, a + b, a - b are all coprime integers.
>
> Any comment about the ccorrectness of the Statement will be
> appreciated.


HINT If d|b then d|a iff d|a-kb

Hence a, b and a-kb, b have the same set of common divisors d,
hence the same *greatest* common divisor, i.e. gcd(a,b) = gcd(a-kb,b).

--Bill Dubuque



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