What are the divisors of 855?

1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855

There is a total of 12 positive divisors.
The sum of these divisors is 1560.
The arithmetic mean is 130.

12 odd divisors

1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855

How to compute the divisors of 855?

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 855 by each of the numbers from 1 to 855 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)

855 / 1 = 855 (the remainder is 0, so 1 is a divisor of 855)
855 / 2 = 427.5 (the remainder is 1, so 2 is not a divisor of 855)
855 / 3 = 285 (the remainder is 0, so 3 is a divisor of 855)
...
855 / 854 = 1.0011709601874 (the remainder is 1, so 854 is not a divisor of 855)
855 / 855 = 1 (the remainder is 0, so 855 is a divisor of 855)

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 855 (i.e. 29.240383034427). 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 $\frac{N}{d} = D$ )

855 / 1 = 855 (the remainder is 0, so 1 and 855 are divisors of 855)
855 / 2 = 427.5 (the remainder is 1, so 2 is not a divisor of 855)
855 / 3 = 285 (the remainder is 0, so 3 and 285 are divisors of 855)
...
855 / 28 = 30.535714285714 (the remainder is 15, so 28 is not a divisor of 855)
855 / 29 = 29.48275862069 (the remainder is 14, so 29 is not a divisor of 855)