What are the divisors of 2099?

1, 2099

There is a total of 2 positive divisors.
The sum of these divisors is 2100.
The arithmetic mean is 1050.

2 odd divisors

1, 2099

How to compute the divisors of 2099?

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 2099 by each of the numbers from 1 to 2099 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)

2099 / 1 = 2099 (the remainder is 0, so 1 is a divisor of 2099)
2099 / 2 = 1049.5 (the remainder is 1, so 2 is not a divisor of 2099)
2099 / 3 = 699.66666666667 (the remainder is 2, so 3 is not a divisor of 2099)
...
2099 / 2098 = 1.0004766444233 (the remainder is 1, so 2098 is not a divisor of 2099)
2099 / 2099 = 1 (the remainder is 0, so 2099 is a divisor of 2099)

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 2099 (i.e. 45.814844755821). 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 $\frac{N}{d} = D$ )

2099 / 1 = 2099 (the remainder is 0, so 1 and 2099 are divisors of 2099)
2099 / 2 = 1049.5 (the remainder is 1, so 2 is not a divisor of 2099)
2099 / 3 = 699.66666666667 (the remainder is 2, so 3 is not a divisor of 2099)
...
2099 / 44 = 47.704545454545 (the remainder is 31, so 44 is not a divisor of 2099)
2099 / 45 = 46.644444444444 (the remainder is 29, so 45 is not a divisor of 2099)