SOLUTION: prove, if x,y ∈ Q then there are integers a,b,c such that x=a/c and y=b/c

Algebra ->  Proofs -> SOLUTION: prove, if x,y ∈ Q then there are integers a,b,c such that x=a/c and y=b/c      Log On


   

Question 1042996: prove, if x,y ∈ Q
then there are integers a,b,c
such that x=a/c and y=b/c
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
Since x and y are rational there exist
integers d and e such that x+=+d%2Fe
and integers f and g such that y+=+f%2Fg
Notice that
x+=+d%2Fe = x+=+%28d%2Fe%29%28g%2Fg%29 = x+=+dg%2Feg
and
y+=+f%2Fg = y+=+%28f%2Fg%29%28e%2Fe%29 = y+=+fe%2Feg
So we have x+=+dg%2Feg and y+=+fe%2Feg
Since the product of two integers is an integer,
we can set c = eg where c is an integer
set a = dg where a is an integer
set b = fe where b is an integer
Substituting we have
x+=+a%2Fc and y+=+b%2Fc