build number not divisible by given number

Question

I am curious if I have following proof correct. Given number $N$ I would like to build a number not divisible by $N$.

1. Let's take $Np$, where p is some integer.
2. Now, take $Q= Np + d$ where $0<d<N$.

$Q$ is not divisible by $N$.

Proof: Assume contrary that $Q$ is divisible by $N$. it means $Q = Nt$. We have:

$$Nt = Np + d$$

$$N (t-p) = d$$

Let's divide both sides on $N$. We have:

$$t-p = d/N$$

but on the left side is integer, and it can't equal the right side which is fraction, unless $N$ divides $d$.

Is this proof correct?

Answer 1

It's not correct, just take $d=0,$ Then $Q$ is always divisible by $N.$ Now if $d>0,$ then it's still not correct, as we can take $p>N$ as the integers are unbounded, and then set $d=N.$ It is correct if $1\leq d<N,$ but this forces us to choose $N\not=1.$ So if $N>1,$ and $1<d<N,$ then for any integer $p$ it follows that $Np+d$ is not divisible by $N.$