If $q$ is a prime $\leq p$, then $q$ divides $p\# − q$

Question

If $q$ is a prime $\leq p$, then $q$ divides $p\# − q$

What does this mean?

I know that it is related to something which I have been studying, but what does $p\# − q$ mean? I am only beginning to learn about this notation.

Answer 1

The notation $p\#$ is sometimes used for the product of all the primes $\le p$.

Please see, for example, this article. For more, search using the key word primorial.

If $q\le p$, then $q$ is one of the primes that got multiplied together to form $p\#$. It follows that $q$ divides $p\#$, and therefore $q$ divides $p\#-q$.

Answer 2

Hint: If André Nicolas' meaning for $p\#$ is correct, and $q$ is a prime such that $q \le p$, do you see why $q | p\#$?

Then the stated result should be clear.