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!

One is the remainder divided by n.  The other is the remainder divided by 2.
The remainders are not the same.  The remainders divided by the divisor are the same.  That is $remainder(\frac {S_n}{n})/n  = remainder(\frac {n + 1}{2})/2$
Think of this.
12/8 has remainder 4. 6/4 has remainder 2.  And 3/2 has remainder 1.
4/8 = 2/4 = 1/2 but 4 does not equal 2 does not equal 1.
