cjquines0 wrote: A polynomial $f(x)$ with integer coefficients is said to be tri-divisible if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible. Write $P(x)=x(x-1)(x-2)Q(x)+ax^2+bx+c$ Get then that $c\equiv a+b+c\equiv a-b+c\pmod 3$ and so $a\equiv b\equiv c\equiv 0\pmod 3$ which obviously is also a sufficient condition. Hence the answer : $\boxed{P(x)=x(x-1)(x-2)Q(x)+3(ax^2+bx+c)}$ where $a,b,c\in\mathbb Z$ and $Q(x)\in\mathbb Z[X]$