A positive integer is divisible by $3$ iff 3 divides the sum of its digits I am having trouble proving the two following questions: 


*

*If $p|N$, $q|N$ and gcd(p,q)=1, then prove that $pq|N$

*If $x$ is non zero positive integer number, then prove that $3|x$ if and only if 3 divides the sum of all digits of $x$.
For both questions I tried to use theorems of discrete mathematics, but I could not find the way to solve them.

$x=a_0+10a_1 + 100a_2 + \cdots+10^{n}a_{n}$.
Let $s$ be the sum of all the digits of $x$, that is $s=a_0 + a_1 + a_2 +\cdots a_n$.
$x-s = 9a_1 + 99a_2 + \cdots(10^{n}-1)a_n$.
Clearly $3\vert x-s$. If $3\vert s$, then $3\vert x$. If $3\vert x$, then $3\vert s$. 
Therefore $3|x$ if and only if $3$ divides the sum of all the digits of $x$.
