I am reading the book Discrete Mathematics by Epp, but I can't understand her proposition 4.7.3 proof (page 210).
The proof claim is: for any integer $a$ and any prime number $p$, if $p$ divides $a$, then $p$ does not divide $a+1$.
Due to divisibility $a = pr$ and $a+1 = ps$ for some integers $r$ and $s$.
It follows that $1 = (a+1) – a$
But this is the part I don't understand. How does Epp come to the step that $(a+1) – a$ equals to $1$?
Best Answer
I think you're overthinking the wrong step here. $(a+1)-a = 1$ is really obvious. All it says is that the difference between one integer and the next integer is $1$. It's the next step that requires thinking: Substitute $pr$ and $ps$ for $a$ and $a+1$, and you get $$ 1 = (a+1)-a = ps-pr = p(s-r) $$ which means that $p\mid 1$, which is impossible for a prime.