What are the divisors of 8100?

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 81, 90, 100, 108, 135, 150, 162, 180, 225, 270, 300, 324, 405, 450, 540, 675, 810, 900, 1350, 1620, 2025, 2700, 4050, 8100

There is a total of 45 positive divisors.
The sum of these divisors is 26257.
The arithmetic mean is 583.48888888889.

30 even divisors

2, 4, 6, 10, 12, 18, 20, 30, 36, 50, 54, 60, 90, 100, 108, 150, 162, 180, 270, 300, 324, 450, 540, 810, 900, 1350, 1620, 2700, 4050, 8100

15 odd divisors

1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025

How to compute the divisors of 8100?

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 8100 by each of the numbers from 1 to 8100 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)

8100 / 1 = 8100 (the remainder is 0, so 1 is a divisor of 8100)
8100 / 2 = 4050 (the remainder is 0, so 2 is a divisor of 8100)
8100 / 3 = 2700 (the remainder is 0, so 3 is a divisor of 8100)
...
8100 / 8099 = 1.0001234720336 (the remainder is 1, so 8099 is not a divisor of 8100)
8100 / 8100 = 1 (the remainder is 0, so 8100 is a divisor of 8100)

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 8100 (i.e. 90). 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 $\frac{N}{d} = D$ )

8100 / 1 = 8100 (the remainder is 0, so 1 and 8100 are divisors of 8100)
8100 / 2 = 4050 (the remainder is 0, so 2 and 4050 are divisors of 8100)
8100 / 3 = 2700 (the remainder is 0, so 3 and 2700 are divisors of 8100)
...
8100 / 89 = 91.011235955056 (the remainder is 1, so 89 is not a divisor of 8100)
8100 / 90 = 90 (the remainder is 0, so 90 and 90 are divisors of 8100)