What are the divisors of 1000?

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

There is a total of 16 positive divisors.
The sum of these divisors is 2340.
The arithmetic mean is 146.25.

12 even divisors

2, 4, 8, 10, 20, 40, 50, 100, 200, 250, 500, 1000

4 odd divisors

1, 5, 25, 125

How to compute the divisors of 1000?

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 1000 by each of the numbers from 1 to 1000 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)

1000 / 1 = 1000 (the remainder is 0, so 1 is a divisor of 1000)
1000 / 2 = 500 (the remainder is 0, so 2 is a divisor of 1000)
1000 / 3 = 333.33333333333 (the remainder is 1, so 3 is not a divisor of 1000)
...
1000 / 999 = 1.001001001001 (the remainder is 1, so 999 is not a divisor of 1000)
1000 / 1000 = 1 (the remainder is 0, so 1000 is a divisor of 1000)

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 1000 (i.e. 31.622776601684). 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 $\frac{N}{d} = D$ )

1000 / 1 = 1000 (the remainder is 0, so 1 and 1000 are divisors of 1000)
1000 / 2 = 500 (the remainder is 0, so 2 and 500 are divisors of 1000)
1000 / 3 = 333.33333333333 (the remainder is 1, so 3 is not a divisor of 1000)
...
1000 / 30 = 33.333333333333 (the remainder is 10, so 30 is not a divisor of 1000)
1000 / 31 = 32.258064516129 (the remainder is 8, so 31 is not a divisor of 1000)