What are the divisors of 8205?

1, 3, 5, 15, 547, 1641, 2735, 8205

There is a total of 8 positive divisors.
The sum of these divisors is 13152.
The arithmetic mean is 1644.

8 odd divisors

1, 3, 5, 15, 547, 1641, 2735, 8205

How to compute the divisors of 8205?

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 8205 by each of the numbers from 1 to 8205 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)

8205 / 1 = 8205 (the remainder is 0, so 1 is a divisor of 8205)
8205 / 2 = 4102.5 (the remainder is 1, so 2 is not a divisor of 8205)
8205 / 3 = 2735 (the remainder is 0, so 3 is a divisor of 8205)
...
8205 / 8204 = 1.0001218917601 (the remainder is 1, so 8204 is not a divisor of 8205)
8205 / 8205 = 1 (the remainder is 0, so 8205 is a divisor of 8205)

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 8205 (i.e. 90.581455055657). 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 $\frac{N}{d} = D$ )

8205 / 1 = 8205 (the remainder is 0, so 1 and 8205 are divisors of 8205)
8205 / 2 = 4102.5 (the remainder is 1, so 2 is not a divisor of 8205)
8205 / 3 = 2735 (the remainder is 0, so 3 and 2735 are divisors of 8205)
...
8205 / 89 = 92.191011235955 (the remainder is 17, so 89 is not a divisor of 8205)
8205 / 90 = 91.166666666667 (the remainder is 15, so 90 is not a divisor of 8205)