What are the divisors of 5105?

1, 5, 1021, 5105

There is a total of 4 positive divisors.
The sum of these divisors is 6132.
The arithmetic mean is 1533.

4 odd divisors

1, 5, 1021, 5105

How to compute the divisors of 5105?

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 5105 by each of the numbers from 1 to 5105 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)

5105 / 1 = 5105 (the remainder is 0, so 1 is a divisor of 5105)
5105 / 2 = 2552.5 (the remainder is 1, so 2 is not a divisor of 5105)
5105 / 3 = 1701.6666666667 (the remainder is 2, so 3 is not a divisor of 5105)
...
5105 / 5104 = 1.0001959247649 (the remainder is 1, so 5104 is not a divisor of 5105)
5105 / 5105 = 1 (the remainder is 0, so 5105 is a divisor of 5105)

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 5105 (i.e. 71.449282711585). 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 $\frac{N}{d} = D$ )

5105 / 1 = 5105 (the remainder is 0, so 1 and 5105 are divisors of 5105)
5105 / 2 = 2552.5 (the remainder is 1, so 2 is not a divisor of 5105)
5105 / 3 = 1701.6666666667 (the remainder is 2, so 3 is not a divisor of 5105)
...
5105 / 70 = 72.928571428571 (the remainder is 65, so 70 is not a divisor of 5105)
5105 / 71 = 71.901408450704 (the remainder is 64, so 71 is not a divisor of 5105)