Question about notation in Hardy & Wright's Introduction to the Theory of Numbers

Question

I've recently started reading about number theory and am fascinated, but just beginning my study of it. I've started reading Hardy & Wright's Introduction to the Theory of Numbers. Near the beginning (bottom of p. $5$ to top of p. $6$ in the $6$th edition) is this passage:

Suppose that

$$2, 3, 5, \ldots, p$$

are the primes up to $p$. Then all numbers up to $p$ are divisible by one of these primes, and therefore, if

$$2 \cdot 3 \cdot 5 \ldots p = q,$$

all of the $p – 1$ numbers

$$q + 2, q + 3, q + 4, \ldots, q + p$$

are composite.

I don't understand what they mean by this line:

$$2 \cdot 3 \cdot 5 \ldots p = q$$

and consequently I can't follow the logic to the end of the sentence.

Answer 1

They authors are defining a number $q$, as the product of all the prime numbers from $2$ up to $p$. For example, if we take $p=7$, then we're setting $q$ equal to $2\times3\times5\times7=210$, and we're noting that each number from $212$ to $217$ is composite, with a factor of $2$, $3$, $5$ or $7$.

Answer 2

It means that $q$ is the product $2\times3\times5\times\cdots\times p$.