What are the divisors of 104?

1, 2, 4, 8, 13, 26, 52, 104

There is a total of 8 positive divisors.
The sum of these divisors is 210.
The arithmetic mean is 26.25.

6 even divisors

2, 4, 8, 26, 52, 104

2 odd divisors

1, 13

How to compute the divisors of 104?

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 104 by each of the numbers from 1 to 104 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)

104 / 1 = 104 (the remainder is 0, so 1 is a divisor of 104)
104 / 2 = 52 (the remainder is 0, so 2 is a divisor of 104)
104 / 3 = 34.666666666667 (the remainder is 2, so 3 is not a divisor of 104)
...
104 / 103 = 1.0097087378641 (the remainder is 1, so 103 is not a divisor of 104)
104 / 104 = 1 (the remainder is 0, so 104 is a divisor of 104)

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 104 (i.e. 10.198039027186). 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 $\frac{N}{d} = D$ )

104 / 1 = 104 (the remainder is 0, so 1 and 104 are divisors of 104)
104 / 2 = 52 (the remainder is 0, so 2 and 52 are divisors of 104)
104 / 3 = 34.666666666667 (the remainder is 2, so 3 is not a divisor of 104)
...
104 / 9 = 11.555555555556 (the remainder is 5, so 9 is not a divisor of 104)
104 / 10 = 10.4 (the remainder is 4, so 10 is not a divisor of 104)