
Re:    Solutions of an equation
Posted:
Mar 20, 2013 5:36 PM


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

