What are the divisors of 208?

1, 2, 4, 8, 13, 16, 26, 52, 104, 208

There is a total of 10 positive divisors.
The sum of these divisors is 434.
The arithmetic mean is 43.4.

8 even divisors

2, 4, 8, 16, 26, 52, 104, 208

2 odd divisors

1, 13

How to compute the divisors of 208?

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 208 by each of the numbers from 1 to 208 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)

208 / 1 = 208 (the remainder is 0, so 1 is a divisor of 208)
208 / 2 = 104 (the remainder is 0, so 2 is a divisor of 208)
208 / 3 = 69.333333333333 (the remainder is 1, so 3 is not a divisor of 208)
...
208 / 207 = 1.0048309178744 (the remainder is 1, so 207 is not a divisor of 208)
208 / 208 = 1 (the remainder is 0, so 208 is a divisor of 208)

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 208 (i.e. 14.422205101856). 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 $\frac{N}{d} = D$ )

208 / 1 = 208 (the remainder is 0, so 1 and 208 are divisors of 208)
208 / 2 = 104 (the remainder is 0, so 2 and 104 are divisors of 208)
208 / 3 = 69.333333333333 (the remainder is 1, so 3 is not a divisor of 208)
...
208 / 13 = 16 (the remainder is 0, so 13 and 16 are divisors of 208)
208 / 14 = 14.857142857143 (the remainder is 12, so 14 is not a divisor of 208)