Question 87
$$
S = \{0, 1, 1+2, 1+2+3, \dots, 1+2+3+\dots+(n-1)\}.
$$
a) If $n$ is a power of $2$, prove that all the elements of $S$ have different remainders modulo $n$.
b) If $n$ is not a power of $2$, prove that there exist two elements of $S$ with the same remainder modulo $n$.
S = \{0, 1, 1+2, 1+2+3, \dots, 1+2+3+\dots+(n-1)\}.
$$
a) If $n$ is a power of $2$, prove that all the elements of $S$ have different remainders modulo $n$.
b) If $n$ is not a power of $2$, prove that there exist two elements of $S$ with the same remainder modulo $n$.