What are the divisors of 2000?

1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000, 2000

There is a total of 20 positive divisors.
The sum of these divisors is 4836.
The arithmetic mean is 241.8.

16 even divisors

2, 4, 8, 10, 16, 20, 40, 50, 80, 100, 200, 250, 400, 500, 1000, 2000

4 odd divisors

1, 5, 25, 125

How to compute the divisors of 2000?

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 2000 by each of the numbers from 1 to 2000 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)

2000 / 1 = 2000 (the remainder is 0, so 1 is a divisor of 2000)
2000 / 2 = 1000 (the remainder is 0, so 2 is a divisor of 2000)
2000 / 3 = 666.66666666667 (the remainder is 2, so 3 is not a divisor of 2000)
...
2000 / 1999 = 1.0005002501251 (the remainder is 1, so 1999 is not a divisor of 2000)
2000 / 2000 = 1 (the remainder is 0, so 2000 is a divisor of 2000)

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 2000 (i.e. 44.721359549996). 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 $\frac{N}{d} = D$ )

2000 / 1 = 2000 (the remainder is 0, so 1 and 2000 are divisors of 2000)
2000 / 2 = 1000 (the remainder is 0, so 2 and 1000 are divisors of 2000)
2000 / 3 = 666.66666666667 (the remainder is 2, so 3 is not a divisor of 2000)
...
2000 / 43 = 46.511627906977 (the remainder is 22, so 43 is not a divisor of 2000)
2000 / 44 = 45.454545454545 (the remainder is 20, so 44 is not a divisor of 2000)