What are the divisors of 529?

1, 23, 529

There is a total of 3 positive divisors.
The sum of these divisors is 553.
The arithmetic mean is 184.33333333333.

3 odd divisors

1, 23, 529

How to compute the divisors of 529?

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 529 by each of the numbers from 1 to 529 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)

529 / 1 = 529 (the remainder is 0, so 1 is a divisor of 529)
529 / 2 = 264.5 (the remainder is 1, so 2 is not a divisor of 529)
529 / 3 = 176.33333333333 (the remainder is 1, so 3 is not a divisor of 529)
...
529 / 528 = 1.0018939393939 (the remainder is 1, so 528 is not a divisor of 529)
529 / 529 = 1 (the remainder is 0, so 529 is a divisor of 529)

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 529 (i.e. 23). 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 $\frac{N}{d} = D$ )

529 / 1 = 529 (the remainder is 0, so 1 and 529 are divisors of 529)
529 / 2 = 264.5 (the remainder is 1, so 2 is not a divisor of 529)
529 / 3 = 176.33333333333 (the remainder is 1, so 3 is not a divisor of 529)
...
529 / 22 = 24.045454545455 (the remainder is 1, so 22 is not a divisor of 529)
529 / 23 = 23 (the remainder is 0, so 23 and 23 are divisors of 529)