What are the divisors of 5239?

1, 13, 31, 169, 403, 5239

There is a total of 6 positive divisors.
The sum of these divisors is 5856.
The arithmetic mean is 976.

6 odd divisors

1, 13, 31, 169, 403, 5239

How to compute the divisors of 5239?

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 5239 by each of the numbers from 1 to 5239 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)

5239 / 1 = 5239 (the remainder is 0, so 1 is a divisor of 5239)
5239 / 2 = 2619.5 (the remainder is 1, so 2 is not a divisor of 5239)
5239 / 3 = 1746.3333333333 (the remainder is 1, so 3 is not a divisor of 5239)
...
5239 / 5238 = 1.000190912562 (the remainder is 1, so 5238 is not a divisor of 5239)
5239 / 5239 = 1 (the remainder is 0, so 5239 is a divisor of 5239)

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 5239 (i.e. 72.38093671679). 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 $\frac{N}{d} = D$ )

5239 / 1 = 5239 (the remainder is 0, so 1 and 5239 are divisors of 5239)
5239 / 2 = 2619.5 (the remainder is 1, so 2 is not a divisor of 5239)
5239 / 3 = 1746.3333333333 (the remainder is 1, so 3 is not a divisor of 5239)
...
5239 / 71 = 73.788732394366 (the remainder is 56, so 71 is not a divisor of 5239)
5239 / 72 = 72.763888888889 (the remainder is 55, so 72 is not a divisor of 5239)