What are the divisors of 811?

1, 811

There is a total of 2 positive divisors.
The sum of these divisors is 812.
The arithmetic mean is 406.

2 odd divisors

1, 811

How to compute the divisors of 811?

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 811 by each of the numbers from 1 to 811 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)

811 / 1 = 811 (the remainder is 0, so 1 is a divisor of 811)
811 / 2 = 405.5 (the remainder is 1, so 2 is not a divisor of 811)
811 / 3 = 270.33333333333 (the remainder is 1, so 3 is not a divisor of 811)
...
811 / 810 = 1.0012345679012 (the remainder is 1, so 810 is not a divisor of 811)
811 / 811 = 1 (the remainder is 0, so 811 is a divisor of 811)

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 811 (i.e. 28.478061731796). 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 $\frac{N}{d} = D$ )

811 / 1 = 811 (the remainder is 0, so 1 and 811 are divisors of 811)
811 / 2 = 405.5 (the remainder is 1, so 2 is not a divisor of 811)
811 / 3 = 270.33333333333 (the remainder is 1, so 3 is not a divisor of 811)
...
811 / 27 = 30.037037037037 (the remainder is 1, so 27 is not a divisor of 811)
811 / 28 = 28.964285714286 (the remainder is 27, so 28 is not a divisor of 811)