What are the divisors of 807?

1, 3, 269, 807

There is a total of 4 positive divisors.
The sum of these divisors is 1080.
The arithmetic mean is 270.

4 odd divisors

1, 3, 269, 807

How to compute the divisors of 807?

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 807 by each of the numbers from 1 to 807 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)

807 / 1 = 807 (the remainder is 0, so 1 is a divisor of 807)
807 / 2 = 403.5 (the remainder is 1, so 2 is not a divisor of 807)
807 / 3 = 269 (the remainder is 0, so 3 is a divisor of 807)
...
807 / 806 = 1.0012406947891 (the remainder is 1, so 806 is not a divisor of 807)
807 / 807 = 1 (the remainder is 0, so 807 is a divisor of 807)

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 807 (i.e. 28.407745422684). 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 $\frac{N}{d} = D$ )

807 / 1 = 807 (the remainder is 0, so 1 and 807 are divisors of 807)
807 / 2 = 403.5 (the remainder is 1, so 2 is not a divisor of 807)
807 / 3 = 269 (the remainder is 0, so 3 and 269 are divisors of 807)
...
807 / 27 = 29.888888888889 (the remainder is 24, so 27 is not a divisor of 807)
807 / 28 = 28.821428571429 (the remainder is 23, so 28 is not a divisor of 807)