How can one prove that if x is an integer greater than 2, then x/(x-1) is a not an integer?

Question

How can one prove that if x is an integer greater than 2, then x/(x-1) is a not integer?

Intuitively I can see this is true but how to prove it?

Answer 1

$$\dfrac x{x-1}=1+\dfrac1{x-1} $$

So, $x-1(\ne0)$ must divide $1\implies x-1=\pm1$

Answer 2

Since one answer has already been given, here is a slightly different way of proceeding (there are tweaks which can be made to fill out the proof, if necessary)

$$\frac x{x-1}-1=\frac 1{x-1}\gt0$$

$$2-\frac x{x-1}=\frac {x-2}{x-1}\gt 0$$

So $\frac x{x-1}$ lies strictly between the consecutive integers $1$and $2$, and the second of the inequalities shows explicitly why the condition $x\gt 2$ comes in.

Answer 3

The function $f(x)=\frac{x}{x-1}$ is decreasing for $x\in (2,+\infty)$ and bounded $1<f(x)<2$.

Indeed, for $2<x_1<x_2$: $$1<\frac{x_2}{x_2-1}<\frac{x_1}{x_1-1}<2.$$