What are the divisors of 831?

1, 3, 277, 831

There is a total of 4 positive divisors.
The sum of these divisors is 1112.
The arithmetic mean is 278.

4 odd divisors

1, 3, 277, 831

How to compute the divisors of 831?

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 831 by each of the numbers from 1 to 831 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)

831 / 1 = 831 (the remainder is 0, so 1 is a divisor of 831)
831 / 2 = 415.5 (the remainder is 1, so 2 is not a divisor of 831)
831 / 3 = 277 (the remainder is 0, so 3 is a divisor of 831)
...
831 / 830 = 1.0012048192771 (the remainder is 1, so 830 is not a divisor of 831)
831 / 831 = 1 (the remainder is 0, so 831 is a divisor of 831)

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 831 (i.e. 28.827070610799). 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 $\frac{N}{d} = D$ )

831 / 1 = 831 (the remainder is 0, so 1 and 831 are divisors of 831)
831 / 2 = 415.5 (the remainder is 1, so 2 is not a divisor of 831)
831 / 3 = 277 (the remainder is 0, so 3 and 277 are divisors of 831)
...
831 / 27 = 30.777777777778 (the remainder is 21, so 27 is not a divisor of 831)
831 / 28 = 29.678571428571 (the remainder is 19, so 28 is not a divisor of 831)