What are the divisors of 519?

1, 3, 173, 519

There is a total of 4 positive divisors.
The sum of these divisors is 696.
The arithmetic mean is 174.

4 odd divisors

1, 3, 173, 519

How to compute the divisors of 519?

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 519 by each of the numbers from 1 to 519 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)

519 / 1 = 519 (the remainder is 0, so 1 is a divisor of 519)
519 / 2 = 259.5 (the remainder is 1, so 2 is not a divisor of 519)
519 / 3 = 173 (the remainder is 0, so 3 is a divisor of 519)
...
519 / 518 = 1.0019305019305 (the remainder is 1, so 518 is not a divisor of 519)
519 / 519 = 1 (the remainder is 0, so 519 is a divisor of 519)

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 519 (i.e. 22.781571499789). 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 $\frac{N}{d} = D$ )

519 / 1 = 519 (the remainder is 0, so 1 and 519 are divisors of 519)
519 / 2 = 259.5 (the remainder is 1, so 2 is not a divisor of 519)
519 / 3 = 173 (the remainder is 0, so 3 and 173 are divisors of 519)
...
519 / 21 = 24.714285714286 (the remainder is 15, so 21 is not a divisor of 519)
519 / 22 = 23.590909090909 (the remainder is 13, so 22 is not a divisor of 519)