I came up with this small doubt in basics while going through the Diophantine equation.
My question is if $d$ divides $a$ then does d divides $ax$ for all real numbers?
For the case where $x$ is an integer it is true but what aboutthe other. It seems like there is nothing wrong with it but if so then it will lead to some absurd results.
If I ask the question in a different way: Do we consider $0.4 = 4*0.1$ as a number divicible by $4$ ?
Divisibility is with respect to a given ring . In the ring of integers 2|6 since 6=2*3 and since 6x=2*3x 2 also divides 6x for any INTEGER x. The Real numbers is not only a ring but a field (a non zero real x has and inverse x^-1 so 1=xx^-1 so x divides 1 so x divides any real number y . y=x(x^-1 y) . The concept of x divides y is therefore trivial and not usually referred to in this way .