What are the divisors of 6000?

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 125, 150, 200, 240, 250, 300, 375, 400, 500, 600, 750, 1000, 1200, 1500, 2000, 3000, 6000

There is a total of 40 positive divisors.
The sum of these divisors is 19344.
The arithmetic mean is 483.6.

32 even divisors

2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 40, 48, 50, 60, 80, 100, 120, 150, 200, 240, 250, 300, 400, 500, 600, 750, 1000, 1200, 1500, 2000, 3000, 6000

8 odd divisors

1, 3, 5, 15, 25, 75, 125, 375

How to compute the divisors of 6000?

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 6000 by each of the numbers from 1 to 6000 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)

6000 / 1 = 6000 (the remainder is 0, so 1 is a divisor of 6000)
6000 / 2 = 3000 (the remainder is 0, so 2 is a divisor of 6000)
6000 / 3 = 2000 (the remainder is 0, so 3 is a divisor of 6000)
...
6000 / 5999 = 1.0001666944491 (the remainder is 1, so 5999 is not a divisor of 6000)
6000 / 6000 = 1 (the remainder is 0, so 6000 is a divisor of 6000)

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 6000 (i.e. 77.459666924148). 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 $\frac{N}{d} = D$ )

6000 / 1 = 6000 (the remainder is 0, so 1 and 6000 are divisors of 6000)
6000 / 2 = 3000 (the remainder is 0, so 2 and 3000 are divisors of 6000)
6000 / 3 = 2000 (the remainder is 0, so 3 and 2000 are divisors of 6000)
...
6000 / 76 = 78.947368421053 (the remainder is 72, so 76 is not a divisor of 6000)
6000 / 77 = 77.922077922078 (the remainder is 71, so 77 is not a divisor of 6000)