What are the divisors of 8001?

1, 3, 7, 9, 21, 63, 127, 381, 889, 1143, 2667, 8001

There is a total of 12 positive divisors.
The sum of these divisors is 13312.
The arithmetic mean is 1109.3333333333.

12 odd divisors

1, 3, 7, 9, 21, 63, 127, 381, 889, 1143, 2667, 8001

How to compute the divisors of 8001?

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 8001 by each of the numbers from 1 to 8001 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)

8001 / 1 = 8001 (the remainder is 0, so 1 is a divisor of 8001)
8001 / 2 = 4000.5 (the remainder is 1, so 2 is not a divisor of 8001)
8001 / 3 = 2667 (the remainder is 0, so 3 is a divisor of 8001)
...
8001 / 8000 = 1.000125 (the remainder is 1, so 8000 is not a divisor of 8001)
8001 / 8001 = 1 (the remainder is 0, so 8001 is a divisor of 8001)

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 8001 (i.e. 89.448309095253). 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 $\frac{N}{d} = D$ )

8001 / 1 = 8001 (the remainder is 0, so 1 and 8001 are divisors of 8001)
8001 / 2 = 4000.5 (the remainder is 1, so 2 is not a divisor of 8001)
8001 / 3 = 2667 (the remainder is 0, so 3 and 2667 are divisors of 8001)
...
8001 / 88 = 90.920454545455 (the remainder is 81, so 88 is not a divisor of 8001)
8001 / 89 = 89.898876404494 (the remainder is 80, so 89 is not a divisor of 8001)