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.
(1)  The hypotheses $p|N$ and $q|N$ give us two integers $m,k$ so that $N=mp$ and $N=kq$.  This implies $mp=kq$ so that $q|mp$.  Now, since $gcd(p,q)=1$ and $q|mp$ we know that $q|m$ (think about why this is true if it's not clear).  Then  $m=sq$ for some integer $s$.  Putting this all together, we have $N=mp=sqp=s(pq)$ so $pq|N$.

(2) Here we have an if and only if statement so you'll have to prove two statements/directions(or both at once):

(i) if $3|x$ then $3$ divides the sum of the digits of $x$

(ii) if $3$ divides the sum of the digits of $x$ then $3$ divides $x$

The strategy is to write the number in base $10$, for example (not a proof):

$1356=1\cdot 10^3+3\cdot 10^2+5\cdot 10^1 + 6\cdot 10^0$

Now $10$ has remainder $1$ under division by $3$ so the remainder of $1356$ under division by $3$ is $1 \cdot 1^3 + 3\cdot 1^2 +5\cdot 1^1 +6\cdot 1^0=1+3+5+6$ which is exactly the sum of the digits.  Then the remainder under division by $3$ of $1356$ and $1+3+5+6$ are the same, and $3|1356$ if and only if $3|1+3+5+6$.  

Try to do this in general for some number $n$ with logical steps following the idea of the example above to write a formal proof.
Hints.

(1) Use the definitions. What does it mean for an integer to divide another? What does it mean for two integers to be relatively prime? Do you know that the $\text{GCD}$ of two integers may be expressed as integral linear combinations of these integers?

(2) Just expand the number out in powers of $10$ in the usual decimal convention. Then note that any positive power of $10$ reduces to $1$ modulo $3.$ Can you see why the rule for divisibility for $3$ and $9$ are similar?