What are the divisors of 5317?

1, 13, 409, 5317

There is a total of 4 positive divisors.
The sum of these divisors is 5740.
The arithmetic mean is 1435.

4 odd divisors

1, 13, 409, 5317

How to compute the divisors of 5317?

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 5317 by each of the numbers from 1 to 5317 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)

5317 / 1 = 5317 (the remainder is 0, so 1 is a divisor of 5317)
5317 / 2 = 2658.5 (the remainder is 1, so 2 is not a divisor of 5317)
5317 / 3 = 1772.3333333333 (the remainder is 1, so 3 is not a divisor of 5317)
...
5317 / 5316 = 1.0001881113619 (the remainder is 1, so 5316 is not a divisor of 5317)
5317 / 5317 = 1 (the remainder is 0, so 5317 is a divisor of 5317)

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 5317 (i.e. 72.917761896537). 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 $\frac{N}{d} = D$ )

5317 / 1 = 5317 (the remainder is 0, so 1 and 5317 are divisors of 5317)
5317 / 2 = 2658.5 (the remainder is 1, so 2 is not a divisor of 5317)
5317 / 3 = 1772.3333333333 (the remainder is 1, so 3 is not a divisor of 5317)
...
5317 / 71 = 74.887323943662 (the remainder is 63, so 71 is not a divisor of 5317)
5317 / 72 = 73.847222222222 (the remainder is 61, so 72 is not a divisor of 5317)