What are the divisors of 1909?

1, 23, 83, 1909

There is a total of 4 positive divisors.
The sum of these divisors is 2016.
The arithmetic mean is 504.

4 odd divisors

1, 23, 83, 1909

How to compute the divisors of 1909?

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 1909 by each of the numbers from 1 to 1909 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)

1909 / 1 = 1909 (the remainder is 0, so 1 is a divisor of 1909)
1909 / 2 = 954.5 (the remainder is 1, so 2 is not a divisor of 1909)
1909 / 3 = 636.33333333333 (the remainder is 1, so 3 is not a divisor of 1909)
...
1909 / 1908 = 1.0005241090147 (the remainder is 1, so 1908 is not a divisor of 1909)
1909 / 1909 = 1 (the remainder is 0, so 1909 is a divisor of 1909)

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 1909 (i.e. 43.692104549907). 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 $\frac{N}{d} = D$ )

1909 / 1 = 1909 (the remainder is 0, so 1 and 1909 are divisors of 1909)
1909 / 2 = 954.5 (the remainder is 1, so 2 is not a divisor of 1909)
1909 / 3 = 636.33333333333 (the remainder is 1, so 3 is not a divisor of 1909)
...
1909 / 42 = 45.452380952381 (the remainder is 19, so 42 is not a divisor of 1909)
1909 / 43 = 44.395348837209 (the remainder is 17, so 43 is not a divisor of 1909)