What are the divisors of 8009?

1, 8009

There is a total of 2 positive divisors.
The sum of these divisors is 8010.
The arithmetic mean is 4005.

2 odd divisors

1, 8009

How to compute the divisors of 8009?

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 8009 by each of the numbers from 1 to 8009 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)

8009 / 1 = 8009 (the remainder is 0, so 1 is a divisor of 8009)
8009 / 2 = 4004.5 (the remainder is 1, so 2 is not a divisor of 8009)
8009 / 3 = 2669.6666666667 (the remainder is 2, so 3 is not a divisor of 8009)
...
8009 / 8008 = 1.0001248751249 (the remainder is 1, so 8008 is not a divisor of 8009)
8009 / 8009 = 1 (the remainder is 0, so 8009 is a divisor of 8009)

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 8009 (i.e. 89.493016487322). 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 $\frac{N}{d} = D$ )

8009 / 1 = 8009 (the remainder is 0, so 1 and 8009 are divisors of 8009)
8009 / 2 = 4004.5 (the remainder is 1, so 2 is not a divisor of 8009)
8009 / 3 = 2669.6666666667 (the remainder is 2, so 3 is not a divisor of 8009)
...
8009 / 88 = 91.011363636364 (the remainder is 1, so 88 is not a divisor of 8009)
8009 / 89 = 89.988764044944 (the remainder is 88, so 89 is not a divisor of 8009)