The product of five consecutive positive integers cannot be the square of an integer 
Prove that the product of five consecutive positive integers cannot be the square of an integer.

I don't understand the book's argument below for why $24r-1$ and $24r+5$ can't be one of the five consecutive numbers. Are they saying that since $24-1$ and $24+5$ aren't perfect squares it can't be so? Also, the argument after that about how $24r+4$ is divisible by $6r+1$ and thus is a perfect square is unclear.
Book's solution:


$24r-1$ and $24r+5$ are also divisible neither by $2$ nor by $3$. So they must also be coprime to the remaining four numbers, and thus must be squares.
But this is impossible, because we already know that $24r+1$ is a square, and two non-zero squares can't differ by $2$ or $4$.
For the second part: $6r+1$ is coprime to $24r,24r+1,24r+2$, and $24r+3$. So it must be a square. Hence $24r+4=4(6r+1)$ is a square. But then the two perfect squares $24r+1$ and $24r+4$ differ by $3$, and the only two squares differing by $3$ are $1$ and $4$. This forces $r=0$, which contradicts $r=k(3k\pm 1)/2$.
