What are the divisors of 8165?

1, 5, 23, 71, 115, 355, 1633, 8165

There is a total of 8 positive divisors.
The sum of these divisors is 10368.
The arithmetic mean is 1296.

8 odd divisors

1, 5, 23, 71, 115, 355, 1633, 8165

How to compute the divisors of 8165?

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 8165 by each of the numbers from 1 to 8165 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)

8165 / 1 = 8165 (the remainder is 0, so 1 is a divisor of 8165)
8165 / 2 = 4082.5 (the remainder is 1, so 2 is not a divisor of 8165)
8165 / 3 = 2721.6666666667 (the remainder is 2, so 3 is not a divisor of 8165)
...
8165 / 8164 = 1.000122488976 (the remainder is 1, so 8164 is not a divisor of 8165)
8165 / 8165 = 1 (the remainder is 0, so 8165 is a divisor of 8165)

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 8165 (i.e. 90.36038955206). 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 $\frac{N}{d} = D$ )

8165 / 1 = 8165 (the remainder is 0, so 1 and 8165 are divisors of 8165)
8165 / 2 = 4082.5 (the remainder is 1, so 2 is not a divisor of 8165)
8165 / 3 = 2721.6666666667 (the remainder is 2, so 3 is not a divisor of 8165)
...
8165 / 89 = 91.741573033708 (the remainder is 66, so 89 is not a divisor of 8165)
8165 / 90 = 90.722222222222 (the remainder is 65, so 90 is not a divisor of 8165)