What are the divisors of 8017?

1, 8017

There is a total of 2 positive divisors.
The sum of these divisors is 8018.
The arithmetic mean is 4009.

2 odd divisors

1, 8017

How to compute the divisors of 8017?

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 8017 by each of the numbers from 1 to 8017 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)

8017 / 1 = 8017 (the remainder is 0, so 1 is a divisor of 8017)
8017 / 2 = 4008.5 (the remainder is 1, so 2 is not a divisor of 8017)
8017 / 3 = 2672.3333333333 (the remainder is 1, so 3 is not a divisor of 8017)
...
8017 / 8016 = 1.000124750499 (the remainder is 1, so 8016 is not a divisor of 8017)
8017 / 8017 = 1 (the remainder is 0, so 8017 is a divisor of 8017)

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 8017 (i.e. 89.537701556383). 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 $\frac{N}{d} = D$ )

8017 / 1 = 8017 (the remainder is 0, so 1 and 8017 are divisors of 8017)
8017 / 2 = 4008.5 (the remainder is 1, so 2 is not a divisor of 8017)
8017 / 3 = 2672.3333333333 (the remainder is 1, so 3 is not a divisor of 8017)
...
8017 / 88 = 91.102272727273 (the remainder is 9, so 88 is not a divisor of 8017)
8017 / 89 = 90.078651685393 (the remainder is 7, so 89 is not a divisor of 8017)