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 $$ a_1b_1^k+a_2b_2^k+\cdots+a_mb_m^k=0 $$ for every $1\leq k \leq n$, the condition, $$ r\mid a_1b_1^r+a_2b_2^r+\cdots+a_mb_m^r $$ also holds.