3. Let $a, b \in \mathbf{Z}, m$ be a positive integer. If dividing $a, b$ by $m$ yields the same remainder, then $a$ and $b$ are said to be congruent modulo $m$, denoted as $a \equiv b(\bmod m)$. Among the following statements:
(1) If $a \equiv b(\bmod m), d$ is a positive divisor of $m$, then $a \equiv b(\bmod d)$;
(2) If $a \equiv b(\bmod m), a \equiv b(\bmod n)$, then $a \equiv b(\bmod m n)$;
(3) If $c$ is a positive integer, $a c \equiv b c(\bmod m)$, then $a \equiv b(\bmod m)$;
(4) Let $f(x)=x^{3}-2 x+5$, if $a \equiv b(\bmod m)$, then $f(a) \equiv f(b)(\bmod m)$.
The correct statements are $\qquad$ . (Write down the sequence numbers of all correct statements)
From $a \equiv b(\bmod m) \Leftrightarrow m \mid(a-b)$, we know (1) is correct, (2) and (3) are incorrect;
If $a \equiv b(\bmod m), c \equiv d(\bmod m)$, then $m \mid c(a-b)+b(c-d)=a c-b d$
$\Rightarrow a c \equiv b d(\bmod m)$, thus $a \equiv b(\bmod m) \Rightarrow a^{n} \equiv b^{n}(\bmod m)$
$\Rightarrow f(a) \equiv f(b)(\bmod m)$, so (4) is correct. Therefore, the answer is (1)(4).