What are the divisors of 255?

1, 3, 5, 15, 17, 51, 85, 255

There is a total of 8 positive divisors.
The sum of these divisors is 432.
The arithmetic mean is 54.

8 odd divisors

1, 3, 5, 15, 17, 51, 85, 255

How to compute the divisors of 255?

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 255 by each of the numbers from 1 to 255 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)

255 / 1 = 255 (the remainder is 0, so 1 is a divisor of 255)
255 / 2 = 127.5 (the remainder is 1, so 2 is not a divisor of 255)
255 / 3 = 85 (the remainder is 0, so 3 is a divisor of 255)
...
255 / 254 = 1.003937007874 (the remainder is 1, so 254 is not a divisor of 255)
255 / 255 = 1 (the remainder is 0, so 255 is a divisor of 255)

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 255 (i.e. 15.968719422671). 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 $\frac{N}{d} = D$ )

255 / 1 = 255 (the remainder is 0, so 1 and 255 are divisors of 255)
255 / 2 = 127.5 (the remainder is 1, so 2 is not a divisor of 255)
255 / 3 = 85 (the remainder is 0, so 3 and 85 are divisors of 255)
...
255 / 14 = 18.214285714286 (the remainder is 3, so 14 is not a divisor of 255)
255 / 15 = 17 (the remainder is 0, so 15 and 17 are divisors of 255)