What are the divisors of 8130?

1, 2, 3, 5, 6, 10, 15, 30, 271, 542, 813, 1355, 1626, 2710, 4065, 8130

There is a total of 16 positive divisors.
The sum of these divisors is 19584.
The arithmetic mean is 1224.

8 even divisors

2, 6, 10, 30, 542, 1626, 2710, 8130

8 odd divisors

1, 3, 5, 15, 271, 813, 1355, 4065

How to compute the divisors of 8130?

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 8130 by each of the numbers from 1 to 8130 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)

8130 / 1 = 8130 (the remainder is 0, so 1 is a divisor of 8130)
8130 / 2 = 4065 (the remainder is 0, so 2 is a divisor of 8130)
8130 / 3 = 2710 (the remainder is 0, so 3 is a divisor of 8130)
...
8130 / 8129 = 1.0001230163612 (the remainder is 1, so 8129 is not a divisor of 8130)
8130 / 8130 = 1 (the remainder is 0, so 8130 is a divisor of 8130)

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 8130 (i.e. 90.166512630799). 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 $\frac{N}{d} = D$ )

8130 / 1 = 8130 (the remainder is 0, so 1 and 8130 are divisors of 8130)
8130 / 2 = 4065 (the remainder is 0, so 2 and 4065 are divisors of 8130)
8130 / 3 = 2710 (the remainder is 0, so 3 and 2710 are divisors of 8130)
...
8130 / 89 = 91.348314606742 (the remainder is 31, so 89 is not a divisor of 8130)
8130 / 90 = 90.333333333333 (the remainder is 30, so 90 is not a divisor of 8130)