Question 51
a) Let $n$ be a composite positive integer. Show that there exist integers $a_1, a_2, \dots, a_n$ whose sum is not divisible by $n$, but $n$ divides (at least) one of the numbers
$$
a_k, \quad a_k + a_{k+1}, \quad \dots, \quad a_k + a_{k+1} + \dots + a_{k+n-1},
$$
for all positive integers $k \le n$; $a_k = a_{k-n}$ if $k \gt n$.
b) Let $p$ be prime and let $a_1, a_2, \dots, a_p$ be integers whose sum is not divisible by $p$. Show that there exists a positive integer $k \le p$ such that $p$ divides none of the numbers
$$
a_k, \quad a_k + a_{k+1}, \quad \dots, \quad a_k + a_{k+1} + \dots + a_{k+p-1},
$$
where $a_k = a_{k-p}$ if $k \gt p$.
$$
a_k, \quad a_k + a_{k+1}, \quad \dots, \quad a_k + a_{k+1} + \dots + a_{k+n-1},
$$
for all positive integers $k \le n$; $a_k = a_{k-n}$ if $k \gt n$.
b) Let $p$ be prime and let $a_1, a_2, \dots, a_p$ be integers whose sum is not divisible by $p$. Show that there exists a positive integer $k \le p$ such that $p$ divides none of the numbers
$$
a_k, \quad a_k + a_{k+1}, \quad \dots, \quad a_k + a_{k+1} + \dots + a_{k+p-1},
$$
where $a_k = a_{k-p}$ if $k \gt p$.