What are the divisors of 517?

1, 11, 47, 517

There is a total of 4 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 144.

4 odd divisors

1, 11, 47, 517

How to compute the divisors of 517?

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 517 by each of the numbers from 1 to 517 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)

517 / 1 = 517 (the remainder is 0, so 1 is a divisor of 517)
517 / 2 = 258.5 (the remainder is 1, so 2 is not a divisor of 517)
517 / 3 = 172.33333333333 (the remainder is 1, so 3 is not a divisor of 517)
...
517 / 516 = 1.0019379844961 (the remainder is 1, so 516 is not a divisor of 517)
517 / 517 = 1 (the remainder is 0, so 517 is a divisor of 517)

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 517 (i.e. 22.737634001804). 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 $\frac{N}{d} = D$ )

517 / 1 = 517 (the remainder is 0, so 1 and 517 are divisors of 517)
517 / 2 = 258.5 (the remainder is 1, so 2 is not a divisor of 517)
517 / 3 = 172.33333333333 (the remainder is 1, so 3 is not a divisor of 517)
...
517 / 21 = 24.619047619048 (the remainder is 13, so 21 is not a divisor of 517)
517 / 22 = 23.5 (the remainder is 11, so 22 is not a divisor of 517)