What are the divisors of 5365?

1, 5, 29, 37, 145, 185, 1073, 5365

There is a total of 8 positive divisors.
The sum of these divisors is 6840.
The arithmetic mean is 855.

8 odd divisors

1, 5, 29, 37, 145, 185, 1073, 5365

How to compute the divisors of 5365?

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 5365 by each of the numbers from 1 to 5365 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)

5365 / 1 = 5365 (the remainder is 0, so 1 is a divisor of 5365)
5365 / 2 = 2682.5 (the remainder is 1, so 2 is not a divisor of 5365)
5365 / 3 = 1788.3333333333 (the remainder is 1, so 3 is not a divisor of 5365)
...
5365 / 5364 = 1.0001864280388 (the remainder is 1, so 5364 is not a divisor of 5365)
5365 / 5365 = 1 (the remainder is 0, so 5365 is a divisor of 5365)

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 5365 (i.e. 73.24616030892). 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 $\frac{N}{d} = D$ )

5365 / 1 = 5365 (the remainder is 0, so 1 and 5365 are divisors of 5365)
5365 / 2 = 2682.5 (the remainder is 1, so 2 is not a divisor of 5365)
5365 / 3 = 1788.3333333333 (the remainder is 1, so 3 is not a divisor of 5365)
...
5365 / 72 = 74.513888888889 (the remainder is 37, so 72 is not a divisor of 5365)
5365 / 73 = 73.493150684932 (the remainder is 36, so 73 is not a divisor of 5365)