From the set of numbers $\left\{2,3,4,...,30\right\}$ a number is selected at random. If it is a composite number it is divided by 5 . If is not composite number then it is divided by 3. The probability that the remainder is zero is

The given set of numbers is $\left\{2,3,4,5,...,30\right\}$

Composite numbers are $4,6,9,10,12,14,15,16,18.20,21,22,24,25,26,27,28,30$

Let $A$ be the event of getting a number which is composite number which are divisible by 5 or a prime number which is divisble by 3

The composite numbers which are divisible by 5 are $\left\{10,15,20,25,30\right\}$

Prime numbers which are divisible by 3 is $\left\{3\right\}$

Hence, $n\left(S\right)=29$ and $n\left(A\right)=5+1=6$

Therefore, 

          $\begin{array}{rcl}P\left(A\right)& =& \frac{n\left(A\right)}{n\left(S\right)}\\ & =& \boxed{\frac{6}{29}}\end{array}$