What are the divisors of 810?

1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810

There is a total of 20 positive divisors.
The sum of these divisors is 2178.
The arithmetic mean is 108.9.

10 even divisors

2, 6, 10, 18, 30, 54, 90, 162, 270, 810

10 odd divisors

1, 3, 5, 9, 15, 27, 45, 81, 135, 405

How to compute the divisors of 810?

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 810 by each of the numbers from 1 to 810 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)

810 / 1 = 810 (the remainder is 0, so 1 is a divisor of 810)
810 / 2 = 405 (the remainder is 0, so 2 is a divisor of 810)
810 / 3 = 270 (the remainder is 0, so 3 is a divisor of 810)
...
810 / 809 = 1.0012360939431 (the remainder is 1, so 809 is not a divisor of 810)
810 / 810 = 1 (the remainder is 0, so 810 is a divisor of 810)

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 810 (i.e. 28.460498941515). 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 $\frac{N}{d} = D$ )

810 / 1 = 810 (the remainder is 0, so 1 and 810 are divisors of 810)
810 / 2 = 405 (the remainder is 0, so 2 and 405 are divisors of 810)
810 / 3 = 270 (the remainder is 0, so 3 and 270 are divisors of 810)
...
810 / 27 = 30 (the remainder is 0, so 27 and 30 are divisors of 810)
810 / 28 = 28.928571428571 (the remainder is 26, so 28 is not a divisor of 810)