5. Given that $a$ is any real number. In the following questions, the number of correct conclusions is ( ).
(1) The solution to the equation $a x=0$ is $x=0$;
(2) The solution to the equation $a x=a$ is $x=1$;
(3) The solution to the equation $a x=1$ is $x=\frac{1}{a}$;
(4) The solution to the equation $|a| x=a$ is $x= \pm 1$.
(A) 0
(B) 1
(C) 2
(D) 3
$5 . A$