TITLE: Why in Sieve of Erastothenes of $N$ number you need to check and cross out numbers up to $\sqrt{N}$? How it's proved?
QUESTION [11 upvotes]: Why in Sieve of Erastothenes of $N$ number you need to check and cross out numbers up to $\sqrt{N}$? How it's proved?
For example if $N = 20$:
with $2$ we cross out:

2 4 6 8 10 12 14 16 18 20

with $3$:

3 9 15

and with $5$ we don't need to check because $10$, $15$ and $20$ are already crossed out and same with others biggers.

REPLY [3 votes]: HINT $\rm\ \ N = ab,\ a\le b\ \Rightarrow\ a^2 \le ab\ \Rightarrow\ a\le \sqrt{ab} = \sqrt{N}\:.$