When searching for primes why is prime * prime > number a case to ignore? That is to check to see if a number is a prime we check if it is divisible by previous primes.
However if the number we are checking is number, and prime * prime > number, where number is the number we are checking then...
we know all primes after this will not be a factor of number.
Why is this conceptually?
// simple code implementing this concept
function findPrimes(N) {
  const primes = [2];
  main: for (let i = 3; i <= N; i += 2) {
    for (let j = 1, prime; prime = primes[j++];) {
      if (prime * prime > i) break; // this line here
      if (i % prime === 0) continue main;
    }
    primes.push(i);
  }
  return primes;
}

 A: This is equivalent to saying that you only need to check if a number $n$ is divisible by numbers between $1,2,\cdots,\sqrt{n}$. This is because if $d>\sqrt{n}$ and $d|n$, then $n/d<\sqrt{n}$, meaning that you would have already found another divisor, implying that checking above $\sqrt{n}$ is redundant. 
Example: Take $n=34$. Then $\sqrt{n}=5.8\ldots$ . So if you want to check if $n$ is prime, you only need to check whether $2,3, 5$ divide $n$. Since $2$ divides 34, it's not prime. Notice that 34/2=17, and even though $17$ divides $34$, you've already accounted for it by checking $2$. 
A: Let the prime be $p$
well $p$ divides $$p,2p,3p,4p,5p,6p,\ldots, (p+1)p,\ldots $$
That is, it divides every natural multiple of itself.
What we're looking for is the least prime factor of a number to eliminate it.
If we get to $p$ in our sieve, all numbers less than it have a smaller prime factor, so those multiples of $p$ would be eliminated by smaller primes. It follows that the square of $p$ is the lowest multiple of $p$ not knocked out. 
If our number hasn't been knocked out by previous primes, it's least prime factor is at least $p$ . But that also implies( by above) it's greater than the square of $p$ . 
In the case it's not, then we've got a contradiction to the implication, given. So it's least prime factor is itself. 
