A misunderstanding in elementary number theory Let $n$ be an even positive integer. $S_n={{n(n+1)}\over {2}}$. I know that the remainder of the euclidian division of $S_n$ by $n$ is $n/2$ since 
$$S_n=n*{n\over 2}+{n\over 2};\;\;\; 0\leq {n\over 2}<n $$ but I want to know what is wrong about the following reasoning:
${S_n\over n}={(n+1)\over 2}$ so the remainder of $S_n$ by $n$ is the same as the remainder of $n+1$ by $2$ and since $n+1$ is odd then this remainder is $1$. Thank you for your help!

$S_n/n$ does not mean "the remainder in the euclidean division of $S_n$ by $n$", but "the fraction whose numerator is $S_n$ and whose denominator is $n$".
$S_n/n = (n+1)/2$ means that those two fractions are equal (or that $2S_n = n(n+1)$ if you want to only talk about integers). It has absolutely nothing to do with the remainders of the respective divisions.
