Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations \[ a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \]for every $1 \leq k \leq n$, the condition \[ r \mid a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \]also holds.