What are the divisors of 865?

1, 5, 173, 865

There is a total of 4 positive divisors.
The sum of these divisors is 1044.
The arithmetic mean is 261.

4 odd divisors

1, 5, 173, 865

How to compute the divisors of 865?

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 865 by each of the numbers from 1 to 865 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)

865 / 1 = 865 (the remainder is 0, so 1 is a divisor of 865)
865 / 2 = 432.5 (the remainder is 1, so 2 is not a divisor of 865)
865 / 3 = 288.33333333333 (the remainder is 1, so 3 is not a divisor of 865)
...
865 / 864 = 1.0011574074074 (the remainder is 1, so 864 is not a divisor of 865)
865 / 865 = 1 (the remainder is 0, so 865 is a divisor of 865)

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 865 (i.e. 29.410882339705). 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 $\frac{N}{d} = D$ )

865 / 1 = 865 (the remainder is 0, so 1 and 865 are divisors of 865)
865 / 2 = 432.5 (the remainder is 1, so 2 is not a divisor of 865)
865 / 3 = 288.33333333333 (the remainder is 1, so 3 is not a divisor of 865)
...
865 / 28 = 30.892857142857 (the remainder is 25, so 28 is not a divisor of 865)
865 / 29 = 29.827586206897 (the remainder is 24, so 29 is not a divisor of 865)