Is There a Contradiction in the Brain Teaser with Positive Integers x and y?

[1+1=1]

if x and y are pos. int. then rx >=y. x is an int. help!

 

[Gokul43201]

What is the question ?

If x, y are positive integers, there can always be found an r such that rx >= y. (Why repeat "x is an integer" ?) Do you want a proof of the above statement ?

 

[Gokul43201]

Assume that y > rx , for all r

Then the set S = {y-rx| r in Z+} consists only of positive numbers. So, S must possesses a least element, say y-mx.

But y-(m+1)x also belongs in S, since m+1 is in Z+ if m is.

y-(m+1)x = y-mx - x < y - mx, since x>0, contrary to our choice of the minimal element - a contradiction !

Hence, the assumption was false.