TITLE: $a,b,n,d\in \mathbb N$. $a,b,d$ are different numbers from the interval $(n^2;n^2+n)$. Prove that it can't be true that $d|ab$.
QUESTION [6 upvotes]: $a,b,n,d\in \mathbb N$. $a,b,d$ are different  numbers from the interval $(n^2;n^2+n)$. Prove that it can't be true that $d\mid ab$.
This is an interesting problem and I don't know how to start, so I'd like to get some help from someone. Thanks.

REPLY [2 votes]: Hint $\ \ $ If $\,\rm\ A_i \equiv a_i\pmod d\ $ then $\rm\ d\mid A_1 A_2\iff d\mid a_1 a_2,\, $ which fails if $\rm\ 0 < |a_i| < \sqrt{d}.\ $ 
Thus we have a simple indivisibility test: for "small"  $\rm\,a_i \equiv A_i,\,$  verify that $\rm\,0 < |a_i| < \sqrt d. $ 
For example,  $\rm\ 30\nmid \color{#c00}{25}\cdot \color{#0a0}{34}\ $ by $\ 0 < \color{#c00}5,\color{#0a0}4 < \sqrt{30}.\, $  Applying this test quickly solves your problem.
Remark $ $  This is a simple example of the general principle that conversion of divisibility problems into the equational language of modular arithmetic often yields great simplifications, since it enables us to reuse our well-honed intuition about integer arithmetic.