What are the divisors of 4105?

1, 5, 821, 4105

There is a total of 4 positive divisors.
The sum of these divisors is 4932.
The arithmetic mean is 1233.

4 odd divisors

1, 5, 821, 4105

How to compute the divisors of 4105?

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 4105 by each of the numbers from 1 to 4105 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)

4105 / 1 = 4105 (the remainder is 0, so 1 is a divisor of 4105)
4105 / 2 = 2052.5 (the remainder is 1, so 2 is not a divisor of 4105)
4105 / 3 = 1368.3333333333 (the remainder is 1, so 3 is not a divisor of 4105)
...
4105 / 4104 = 1.0002436647173 (the remainder is 1, so 4104 is not a divisor of 4105)
4105 / 4105 = 1 (the remainder is 0, so 4105 is a divisor of 4105)

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 4105 (i.e. 64.070273918565). 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 $\frac{N}{d} = D$ )

4105 / 1 = 4105 (the remainder is 0, so 1 and 4105 are divisors of 4105)
4105 / 2 = 2052.5 (the remainder is 1, so 2 is not a divisor of 4105)
4105 / 3 = 1368.3333333333 (the remainder is 1, so 3 is not a divisor of 4105)
...
4105 / 63 = 65.15873015873 (the remainder is 10, so 63 is not a divisor of 4105)
4105 / 64 = 64.140625 (the remainder is 9, so 64 is not a divisor of 4105)