What are the divisors of 205?

1, 5, 41, 205

There is a total of 4 positive divisors.
The sum of these divisors is 252.
The arithmetic mean is 63.

4 odd divisors

1, 5, 41, 205

How to compute the divisors of 205?

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 205 by each of the numbers from 1 to 205 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)

205 / 1 = 205 (the remainder is 0, so 1 is a divisor of 205)
205 / 2 = 102.5 (the remainder is 1, so 2 is not a divisor of 205)
205 / 3 = 68.333333333333 (the remainder is 1, so 3 is not a divisor of 205)
...
205 / 204 = 1.0049019607843 (the remainder is 1, so 204 is not a divisor of 205)
205 / 205 = 1 (the remainder is 0, so 205 is a divisor of 205)

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 205 (i.e. 14.317821063276). 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 $\frac{N}{d} = D$ )

205 / 1 = 205 (the remainder is 0, so 1 and 205 are divisors of 205)
205 / 2 = 102.5 (the remainder is 1, so 2 is not a divisor of 205)
205 / 3 = 68.333333333333 (the remainder is 1, so 3 is not a divisor of 205)
...
205 / 13 = 15.769230769231 (the remainder is 10, so 13 is not a divisor of 205)
205 / 14 = 14.642857142857 (the remainder is 9, so 14 is not a divisor of 205)