For all positive integers $n$ less than $30$, let
$a_n=\{\begin{array}{ll}2,& \text{if }n\text{ is divisible by }3\text{ and }5;\\ 3,& \text{if }n\text{ is divisible by }5\text{ and }2;\\ 5,& \text{if }n\text{ is divisible by }2\text{ and }3;\\ 0,& \text{otherwise}.\end{array}$

Calculate ${\sum }_{n=1}^{29}{a}_n$.