What are the divisors of 561?

1, 3, 11, 17, 33, 51, 187, 561

There is a total of 8 positive divisors.
The sum of these divisors is 864.
The arithmetic mean is 108.

8 odd divisors

1, 3, 11, 17, 33, 51, 187, 561

How to compute the divisors of 561?

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 561 by each of the numbers from 1 to 561 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)

561 / 1 = 561 (the remainder is 0, so 1 is a divisor of 561)
561 / 2 = 280.5 (the remainder is 1, so 2 is not a divisor of 561)
561 / 3 = 187 (the remainder is 0, so 3 is a divisor of 561)
...
561 / 560 = 1.0017857142857 (the remainder is 1, so 560 is not a divisor of 561)
561 / 561 = 1 (the remainder is 0, so 561 is a divisor of 561)

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 561 (i.e. 23.685438564654). 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 $\frac{N}{d} = D$ )

561 / 1 = 561 (the remainder is 0, so 1 and 561 are divisors of 561)
561 / 2 = 280.5 (the remainder is 1, so 2 is not a divisor of 561)
561 / 3 = 187 (the remainder is 0, so 3 and 187 are divisors of 561)
...
561 / 22 = 25.5 (the remainder is 11, so 22 is not a divisor of 561)
561 / 23 = 24.391304347826 (the remainder is 9, so 23 is not a divisor of 561)