What are the divisors of 8005?

1, 5, 1601, 8005

There is a total of 4 positive divisors.
The sum of these divisors is 9612.
The arithmetic mean is 2403.

4 odd divisors

1, 5, 1601, 8005

How to compute the divisors of 8005?

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 8005 by each of the numbers from 1 to 8005 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)

8005 / 1 = 8005 (the remainder is 0, so 1 is a divisor of 8005)
8005 / 2 = 4002.5 (the remainder is 1, so 2 is not a divisor of 8005)
8005 / 3 = 2668.3333333333 (the remainder is 1, so 3 is not a divisor of 8005)
...
8005 / 8004 = 1.0001249375312 (the remainder is 1, so 8004 is not a divisor of 8005)
8005 / 8005 = 1 (the remainder is 0, so 8005 is a divisor of 8005)

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 8005 (i.e. 89.470665583754). 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 $\frac{N}{d} = D$ )

8005 / 1 = 8005 (the remainder is 0, so 1 and 8005 are divisors of 8005)
8005 / 2 = 4002.5 (the remainder is 1, so 2 is not a divisor of 8005)
8005 / 3 = 2668.3333333333 (the remainder is 1, so 3 is not a divisor of 8005)
...
8005 / 88 = 90.965909090909 (the remainder is 85, so 88 is not a divisor of 8005)
8005 / 89 = 89.943820224719 (the remainder is 84, so 89 is not a divisor of 8005)