What are the divisors of 8145?

1, 3, 5, 9, 15, 45, 181, 543, 905, 1629, 2715, 8145

There is a total of 12 positive divisors.
The sum of these divisors is 14196.
The arithmetic mean is 1183.

12 odd divisors

1, 3, 5, 9, 15, 45, 181, 543, 905, 1629, 2715, 8145

How to compute the divisors of 8145?

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 8145 by each of the numbers from 1 to 8145 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)

8145 / 1 = 8145 (the remainder is 0, so 1 is a divisor of 8145)
8145 / 2 = 4072.5 (the remainder is 1, so 2 is not a divisor of 8145)
8145 / 3 = 2715 (the remainder is 0, so 3 is a divisor of 8145)
...
8145 / 8144 = 1.0001227897839 (the remainder is 1, so 8144 is not a divisor of 8145)
8145 / 8145 = 1 (the remainder is 0, so 8145 is a divisor of 8145)

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 8145 (i.e. 90.249653738948). 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 $\frac{N}{d} = D$ )

8145 / 1 = 8145 (the remainder is 0, so 1 and 8145 are divisors of 8145)
8145 / 2 = 4072.5 (the remainder is 1, so 2 is not a divisor of 8145)
8145 / 3 = 2715 (the remainder is 0, so 3 and 2715 are divisors of 8145)
...
8145 / 89 = 91.516853932584 (the remainder is 46, so 89 is not a divisor of 8145)
8145 / 90 = 90.5 (the remainder is 45, so 90 is not a divisor of 8145)