Is there a prime number $p$ such that : $p\bmod n =1 $ for $n <10$? and if yes , Are there infinity of them?

Question

I want to know set of primes such that satisfies the below question :

Question: Is there a prime number $p$ such that : $p\equiv 1\bmod n $ for all $n <10$ ? And if yes, are there infinity of them ?

For example : $p\bmod 1 =1 ,p\bmod 2 =1,p\bmod 3 =1,\cdots$

Edit: I have added a question includes wether there are infinity of them without changing the meaning of the question

Answer 1

Yes, a number is congruent to $1$ mod each of $1,2,3,4,...,10$ if and only if it is $1$ mod $2520$. In fact $2521$ is prime so this will do.