To test whether a number is prime or not, why do we have to test whether it is divisible only up to the square root of that number?
If a number n
is not a prime, it can be factored into two factors a
and b
:
n = a * b
Now a
and b
can't be both greater than the square root of n
, since then the product a * b
would be greater than sqrt(n) * sqrt(n) = n
. So in any factorization of n
, at least one of the factors must be smaller than the square root of n
, and if we can't find any factors less than or equal to the square root, n
must be a prime.