What are the divisors of 8011?

1, 8011

There is a total of 2 positive divisors.
The sum of these divisors is 8012.
The arithmetic mean is 4006.

2 odd divisors

1, 8011

How to compute the divisors of 8011?

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 8011 by each of the numbers from 1 to 8011 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)

8011 / 1 = 8011 (the remainder is 0, so 1 is a divisor of 8011)
8011 / 2 = 4005.5 (the remainder is 1, so 2 is not a divisor of 8011)
8011 / 3 = 2670.3333333333 (the remainder is 1, so 3 is not a divisor of 8011)
...
8011 / 8010 = 1.0001248439451 (the remainder is 1, so 8010 is not a divisor of 8011)
8011 / 8011 = 1 (the remainder is 0, so 8011 is a divisor of 8011)

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 8011 (i.e. 89.504189846063). 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 $\frac{N}{d} = D$ )

8011 / 1 = 8011 (the remainder is 0, so 1 and 8011 are divisors of 8011)
8011 / 2 = 4005.5 (the remainder is 1, so 2 is not a divisor of 8011)
8011 / 3 = 2670.3333333333 (the remainder is 1, so 3 is not a divisor of 8011)
...
8011 / 88 = 91.034090909091 (the remainder is 3, so 88 is not a divisor of 8011)
8011 / 89 = 90.011235955056 (the remainder is 1, so 89 is not a divisor of 8011)