What are the divisors of 8181?

1, 3, 9, 27, 81, 101, 303, 909, 2727, 8181

There is a total of 10 positive divisors.
The sum of these divisors is 12342.
The arithmetic mean is 1234.2.

10 odd divisors

1, 3, 9, 27, 81, 101, 303, 909, 2727, 8181

How to compute the divisors of 8181?

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 8181 by each of the numbers from 1 to 8181 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)

8181 / 1 = 8181 (the remainder is 0, so 1 is a divisor of 8181)
8181 / 2 = 4090.5 (the remainder is 1, so 2 is not a divisor of 8181)
8181 / 3 = 2727 (the remainder is 0, so 3 is a divisor of 8181)
...
8181 / 8180 = 1.0001222493888 (the remainder is 1, so 8180 is not a divisor of 8181)
8181 / 8181 = 1 (the remainder is 0, so 8181 is a divisor of 8181)

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 8181 (i.e. 90.448880590088). 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 $\frac{N}{d} = D$ )

8181 / 1 = 8181 (the remainder is 0, so 1 and 8181 are divisors of 8181)
8181 / 2 = 4090.5 (the remainder is 1, so 2 is not a divisor of 8181)
8181 / 3 = 2727 (the remainder is 0, so 3 and 2727 are divisors of 8181)
...
8181 / 89 = 91.921348314607 (the remainder is 82, so 89 is not a divisor of 8181)
8181 / 90 = 90.9 (the remainder is 81, so 90 is not a divisor of 8181)