What are the divisors of 641?

1, 641

There is a total of 2 positive divisors.
The sum of these divisors is 642.
The arithmetic mean is 321.

2 odd divisors

1, 641

How to compute the divisors of 641?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 641 by each of the numbers from 1 to 641 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

641 / 1 = 641 (the remainder is 0, so 1 is a divisor of 641)
641 / 2 = 320.5 (the remainder is 1, so 2 is not a divisor of 641)
641 / 3 = 213.66666666667 (the remainder is 2, so 3 is not a divisor of 641)
...
641 / 640 = 1.0015625 (the remainder is 1, so 640 is not a divisor of 641)
641 / 641 = 1 (the remainder is 0, so 641 is a divisor of 641)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 641 (i.e. 25.317977802344). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

641 / 1 = 641 (the remainder is 0, so 1 and 641 are divisors of 641)
641 / 2 = 320.5 (the remainder is 1, so 2 is not a divisor of 641)
641 / 3 = 213.66666666667 (the remainder is 2, so 3 is not a divisor of 641)
...
641 / 24 = 26.708333333333 (the remainder is 17, so 24 is not a divisor of 641)
641 / 25 = 25.64 (the remainder is 16, so 25 is not a divisor of 641)