Is my method correct to answering this question? Is there a quicker method to solve this question? 
What is the smallest number that can be written as the sum of three, four and five consecutive numbers?

I encountered this question while doing my Math summer homework. I have tried to make progress on this question.
Sum of three consecutive numbers = $x + x+1 + x+2 = 3x+3$
Sum of four consecutive integers = $x + x+1 + x+2 + x+3 = 4x+6$
Sum of five consecutive integers = $x + x+1 + x+2 + x+3 + x+4 = 5x+10$
The number must be the lowest common multiple of $3x+3$ , $4x+6$ and $5x+10$, which is $60x + 30$.
Substituting $x = 0$ gives us the lowest positive, non-zero and whole number, which is $30$.
$$30 = 9 + 10 + 11\\
30 = 6 + 7 + 8 + 9\\
30 = 4 + 5 + 6 + 7 + 8$$
Is my answer correct? If not, where in my method have I produced an error? Is there an ever quicker method to solve this question?

If we are talking about a non-negative integer, I have another method to propose.


Sum of three consecutive numbers = $++1++2=3+3$


Sum of four consecutive integers = $++1++2++3=4+6$


Sum of five consecutive integers = $++1++2++3++4=5+10$

Another way of formulating a solution from this step can be as follows:
$y = 3x_1 + 3 = 4x_2+6 = 5x_3+10$
$y= 3x_1+3\equiv 0 \pmod{3}$
$y= 4x_2+6\equiv 2 \pmod{4}$
$y= 5x_3+10\equiv 0 \pmod{5}$
$y$ is divisible by $5$ and $3$ which gives us $y=15k$
The remainder left from dividing $y$ by $4$ is $2$. $y=15$ doesn't satisfy this condition, but $y=30$ does.
