What are the divisors of 5165?

1, 5, 1033, 5165

There is a total of 4 positive divisors.
The sum of these divisors is 6204.
The arithmetic mean is 1551.

4 odd divisors

1, 5, 1033, 5165

How to compute the divisors of 5165?

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 5165 by each of the numbers from 1 to 5165 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)

5165 / 1 = 5165 (the remainder is 0, so 1 is a divisor of 5165)
5165 / 2 = 2582.5 (the remainder is 1, so 2 is not a divisor of 5165)
5165 / 3 = 1721.6666666667 (the remainder is 2, so 3 is not a divisor of 5165)
...
5165 / 5164 = 1.0001936483346 (the remainder is 1, so 5164 is not a divisor of 5165)
5165 / 5165 = 1 (the remainder is 0, so 5165 is a divisor of 5165)

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 5165 (i.e. 71.867934435324). 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 $\frac{N}{d} = D$ )

5165 / 1 = 5165 (the remainder is 0, so 1 and 5165 are divisors of 5165)
5165 / 2 = 2582.5 (the remainder is 1, so 2 is not a divisor of 5165)
5165 / 3 = 1721.6666666667 (the remainder is 2, so 3 is not a divisor of 5165)
...
5165 / 70 = 73.785714285714 (the remainder is 55, so 70 is not a divisor of 5165)
5165 / 71 = 72.746478873239 (the remainder is 53, so 71 is not a divisor of 5165)