What are the divisors of 631?

1, 631

There is a total of 2 positive divisors.
The sum of these divisors is 632.
The arithmetic mean is 316.

2 odd divisors

1, 631

How to compute the divisors of 631?

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 631 by each of the numbers from 1 to 631 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)

631 / 1 = 631 (the remainder is 0, so 1 is a divisor of 631)
631 / 2 = 315.5 (the remainder is 1, so 2 is not a divisor of 631)
631 / 3 = 210.33333333333 (the remainder is 1, so 3 is not a divisor of 631)
...
631 / 630 = 1.0015873015873 (the remainder is 1, so 630 is not a divisor of 631)
631 / 631 = 1 (the remainder is 0, so 631 is a divisor of 631)

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 631 (i.e. 25.119713374161). 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 $\frac{N}{d} = D$ )

631 / 1 = 631 (the remainder is 0, so 1 and 631 are divisors of 631)
631 / 2 = 315.5 (the remainder is 1, so 2 is not a divisor of 631)
631 / 3 = 210.33333333333 (the remainder is 1, so 3 is not a divisor of 631)
...
631 / 24 = 26.291666666667 (the remainder is 7, so 24 is not a divisor of 631)
631 / 25 = 25.24 (the remainder is 6, so 25 is not a divisor of 631)