What are the divisors of 5321?

1, 17, 313, 5321

There is a total of 4 positive divisors.
The sum of these divisors is 5652.
The arithmetic mean is 1413.

4 odd divisors

1, 17, 313, 5321

How to compute the divisors of 5321?

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 5321 by each of the numbers from 1 to 5321 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)

5321 / 1 = 5321 (the remainder is 0, so 1 is a divisor of 5321)
5321 / 2 = 2660.5 (the remainder is 1, so 2 is not a divisor of 5321)
5321 / 3 = 1773.6666666667 (the remainder is 2, so 3 is not a divisor of 5321)
...
5321 / 5320 = 1.0001879699248 (the remainder is 1, so 5320 is not a divisor of 5321)
5321 / 5321 = 1 (the remainder is 0, so 5321 is a divisor of 5321)

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 5321 (i.e. 72.945184899348). 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 $\frac{N}{d} = D$ )

5321 / 1 = 5321 (the remainder is 0, so 1 and 5321 are divisors of 5321)
5321 / 2 = 2660.5 (the remainder is 1, so 2 is not a divisor of 5321)
5321 / 3 = 1773.6666666667 (the remainder is 2, so 3 is not a divisor of 5321)
...
5321 / 71 = 74.943661971831 (the remainder is 67, so 71 is not a divisor of 5321)
5321 / 72 = 73.902777777778 (the remainder is 65, so 72 is not a divisor of 5321)