What are the divisors of 800?

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800

There is a total of 18 positive divisors.
The sum of these divisors is 1953.
The arithmetic mean is 108.5.

15 even divisors

2, 4, 8, 10, 16, 20, 32, 40, 50, 80, 100, 160, 200, 400, 800

3 odd divisors

1, 5, 25

How to compute the divisors of 800?

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 800 by each of the numbers from 1 to 800 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)

800 / 1 = 800 (the remainder is 0, so 1 is a divisor of 800)
800 / 2 = 400 (the remainder is 0, so 2 is a divisor of 800)
800 / 3 = 266.66666666667 (the remainder is 2, so 3 is not a divisor of 800)
...
800 / 799 = 1.0012515644556 (the remainder is 1, so 799 is not a divisor of 800)
800 / 800 = 1 (the remainder is 0, so 800 is a divisor of 800)

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 800 (i.e. 28.284271247462). 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 $\frac{N}{d} = D$ )

800 / 1 = 800 (the remainder is 0, so 1 and 800 are divisors of 800)
800 / 2 = 400 (the remainder is 0, so 2 and 400 are divisors of 800)
800 / 3 = 266.66666666667 (the remainder is 2, so 3 is not a divisor of 800)
...
800 / 27 = 29.62962962963 (the remainder is 17, so 27 is not a divisor of 800)
800 / 28 = 28.571428571429 (the remainder is 16, so 28 is not a divisor of 800)