
In the last bullet, it says l must be even and provides an explanation. I don't understand the explanation, however. Why does it have to be even?

In the last bullet, it says l must be even and provides an explanation. I don't understand the explanation, however. Why does it have to be even?
I don't really follow the argument given. Here's a correct argument...
If $a^2 = 2l$, then $a^2$ is divisible by 2. But 2 is prime, so $a$ must be divisible by 2 too - say $a = 2b$. Then $(2b)^2 = 2l$, i.e. $2b^2 = l$, and so $l$ is divisible by 2.
They mean that in order to have $a=\sqrt{2}\sqrt{l}$ an integer, we must have that $l$ is an even number, say $2r$. In this case $a=\sqrt{2}\sqrt{2r}=2\sqrt{r}$. If $l$ is odd, we can not spilt of the $\sqrt{2}$ from $\sqrt{l}$, thus the $\sqrt{2}$ in $a$ remains, and $a$ will be irrational.