What are the divisors of 523?

1, 523

There is a total of 2 positive divisors.
The sum of these divisors is 524.
The arithmetic mean is 262.

2 odd divisors

1, 523

How to compute the divisors of 523?

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 523 by each of the numbers from 1 to 523 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)

523 / 1 = 523 (the remainder is 0, so 1 is a divisor of 523)
523 / 2 = 261.5 (the remainder is 1, so 2 is not a divisor of 523)
523 / 3 = 174.33333333333 (the remainder is 1, so 3 is not a divisor of 523)
...
523 / 522 = 1.0019157088123 (the remainder is 1, so 522 is not a divisor of 523)
523 / 523 = 1 (the remainder is 0, so 523 is a divisor of 523)

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 523 (i.e. 22.869193252059). 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 $\frac{N}{d} = D$ )

523 / 1 = 523 (the remainder is 0, so 1 and 523 are divisors of 523)
523 / 2 = 261.5 (the remainder is 1, so 2 is not a divisor of 523)
523 / 3 = 174.33333333333 (the remainder is 1, so 3 is not a divisor of 523)
...
523 / 21 = 24.904761904762 (the remainder is 19, so 21 is not a divisor of 523)
523 / 22 = 23.772727272727 (the remainder is 17, so 22 is not a divisor of 523)