Theorem. Let $f(X) := a_nX^n+\ldots+a_1X+a_0$ be a polynomial in $X$ with integer coefficients $a_i$ and $m$ a positive integer. If the congruence

(1)

Proof. There is an integer $x_0$ such that (1) is satisfied by $$x \;:=\; x_0\!+\!1,\,x_0\!+\!2,\,\ldots,\,x_0\!+\!m.$$ But these values form a complete residue system modulo $m$ . Thus, if $x$ is an arbitrary integer, one has $$x \;\equiv\; x_0\!+\!r \pmod{m} \quad\mbox{where}\;\; 1\leqq r \leqq m.$$ This implies $$a_ix^i \;\equiv\; a_i(x_0\!+\!r)^i \pmod{m} \quad\mbox{for}\;\; i = 0,\,1,\,\ldots,\,n$$ and consequently $$\underbrace{\sum_{i=0}^na_ix^i}_{f(x)} \;\equiv\; \sum_{i=0}^na_i(x_0\!+\!r)^i \;=\; f(x_0\!+\!r) \;\equiv\; 0 \pmod{m}.$$ Accordingly, (1) is true for any integer $x$ , Q.E.D.

Note. Though the congruence (1) is identical, it need not be a question of a formal congruence

(2)