Can every prime (greater than $5$) be expressed in the form $30m+n$? Let
$$
T = \{ 1, 7 , 11 , 13 , 17 , 19 , 23, 29 \}
$$
Can every prime $p > 5$ be expressed in the form $30m+n$, where $m\geq 0$ and $$n is a element of set $T$?

For example 
Prime number $37$ is expressed as $30\times 1+7$ where $7$ is a element of set.
I did check untill $1000$ to find counterexample for this statement. But i can't find it.
If it is true, please help me to prove it.

Yes, prime $\,p>5\,\Rightarrow\,p\,$ is coprime to $\,2,3,5\,$ so $\,p\,$ is coprime to  $\,2\cdot 3\cdot 5 = 30,\,$ thus so too is its remainder $\ n = (p\bmod 30),\,$ hence $\,n\,$ lies in one of the $\,\phi(3) = 8\,$ residue classes that you listed.
Remark $\ 30$ is the largest integer with this property, i.e. every positive integer $> 30$ has at least one composite totative. A proof is in this answer.
