为什么我们要检查质数的平方根以确定它是否为质数? [英] Why do we check up to the square root of a prime number to determine if it is prime?
问题描述
要测试一个数字是否为质数,为什么我们必须测试该数字是否只能整除该数字的平方根?
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?
推荐答案
如果数字 n
不是质数,则可以将其分为两个因素 a
和 b
:
If a number n
is not a prime, it can be factored into two factors a
and b
:
n = a * b
现在 a
和 b
不能都大于 n
的平方根,因为从那以后乘积 a * b
将大于 sqrt(n)* sqrt(n)= n
。因此,在 n
的任何因式分解中,至少一个因子必须小于 n
的平方根,并且如果找不到小于或等于平方根的因子,则 n
必须是质数。
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.
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