What are the divisors of 5399?

1, 5399

There is a total of 2 positive divisors.
The sum of these divisors is 5400.
The arithmetic mean is 2700.

2 odd divisors

1, 5399

How to compute the divisors of 5399?

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 5399 by each of the numbers from 1 to 5399 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)

5399 / 1 = 5399 (the remainder is 0, so 1 is a divisor of 5399)
5399 / 2 = 2699.5 (the remainder is 1, so 2 is not a divisor of 5399)
5399 / 3 = 1799.6666666667 (the remainder is 2, so 3 is not a divisor of 5399)
...
5399 / 5398 = 1.0001852537977 (the remainder is 1, so 5398 is not a divisor of 5399)
5399 / 5399 = 1 (the remainder is 0, so 5399 is a divisor of 5399)

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 5399 (i.e. 73.477887830285). 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 $\frac{N}{d} = D$ )

5399 / 1 = 5399 (the remainder is 0, so 1 and 5399 are divisors of 5399)
5399 / 2 = 2699.5 (the remainder is 1, so 2 is not a divisor of 5399)
5399 / 3 = 1799.6666666667 (the remainder is 2, so 3 is not a divisor of 5399)
...
5399 / 72 = 74.986111111111 (the remainder is 71, so 72 is not a divisor of 5399)
5399 / 73 = 73.958904109589 (the remainder is 70, so 73 is not a divisor of 5399)