What are the divisors of 2105?

1, 5, 421, 2105

There is a total of 4 positive divisors.
The sum of these divisors is 2532.
The arithmetic mean is 633.

4 odd divisors

1, 5, 421, 2105

How to compute the divisors of 2105?

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 2105 by each of the numbers from 1 to 2105 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)

2105 / 1 = 2105 (the remainder is 0, so 1 is a divisor of 2105)
2105 / 2 = 1052.5 (the remainder is 1, so 2 is not a divisor of 2105)
2105 / 3 = 701.66666666667 (the remainder is 2, so 3 is not a divisor of 2105)
...
2105 / 2104 = 1.0004752851711 (the remainder is 1, so 2104 is not a divisor of 2105)
2105 / 2105 = 1 (the remainder is 0, so 2105 is a divisor of 2105)

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 2105 (i.e. 45.880278987818). 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 $\frac{N}{d} = D$ )

2105 / 1 = 2105 (the remainder is 0, so 1 and 2105 are divisors of 2105)
2105 / 2 = 1052.5 (the remainder is 1, so 2 is not a divisor of 2105)
2105 / 3 = 701.66666666667 (the remainder is 2, so 3 is not a divisor of 2105)
...
2105 / 44 = 47.840909090909 (the remainder is 37, so 44 is not a divisor of 2105)
2105 / 45 = 46.777777777778 (the remainder is 35, so 45 is not a divisor of 2105)