What are the divisors of 8?

1, 2, 4, 8

There is a total of 4 positive divisors.
The sum of these divisors is 15.
The arithmetic mean is 3.75.

3 even divisors

2, 4, 8

1 odd divisors

How to compute the divisors of 8?

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 8 by each of the numbers from 1 to 8 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)

8 / 1 = 8 (the remainder is 0, so 1 is a divisor of 8)
8 / 2 = 4 (the remainder is 0, so 2 is a divisor of 8)
8 / 3 = 2.6666666666667 (the remainder is 2, so 3 is not a divisor of 8)
...
8 / 4 = 2 (the remainder is 0, so 4 is a divisor of 8)
8 / 5 = 1.6 (the remainder is 3, so 5 is not a divisor of 8)
8 / 6 = 1.3333333333333 (the remainder is 2, so 6 is not a divisor of 8)
8 / 7 = 1.1428571428571 (the remainder is 1, so 7 is not a divisor of 8)
8 / 8 = 1 (the remainder is 0, so 8 is a divisor of 8)

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 8 (i.e. 2.8284271247462). 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 $\frac{N}{d} = D$ )

8 / 1 = 8 (the remainder is 0, so 1 and 8 are divisors of 8)
8 / 2 = 4 (the remainder is 0, so 2 and 4 are divisors of 8)