Date: Mar 20, 2013 5:36 PM
Author: James Waldby
Subject: Re: ----- ----- -- Solutions of an equation
On Wed, 20 Mar 2013 07:03:52 -0700, Deep wrote:

> Consider (1) below under the given conditions:

>

> XY = AB (1)

>

> Conditions: X, Y, A, B are odd integers each > 1.

> The symbol (m, n) = 1 means m, n are coprime odd integers each > 1. Example m = 3, n = 35.

> (X, Y) = 1, (A, B) = 1

>

> Statement:: Given (1) one gets:

>

> X = A and Y = B (2) or

>

> X = a and Y = b (3) where AB = ab, (a, b) = 1

>

> Any valid comment upon the correctness of the Statement will be appreciated.

Neither (2) nor (3) necessarily follows from (1) given (X,Y)=1 and (A,B)=1.

For example, let X = 3*5, Y = 7*11, A = 3*7, B = 5*11.

Clearly (2) does not hold.

For (3), we have AB = 3*5*7*11 so can take a = 3*7, b = 5*11 and have

neither X=a nor Y=B, so (3) doesn't hold.

However, if you rephrase (3) as

"There exist a,b such that (a,b)=1, A*B = a*b, X=a, Y=b"

then (3) follows trivially from (1) given (X,Y)=1 and (A,B)=1.

--

jiw