What are the divisors of 2039?

1, 2039

There is a total of 2 positive divisors.
The sum of these divisors is 2040.
The arithmetic mean is 1020.

2 odd divisors

1, 2039

How to compute the divisors of 2039?

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 2039 by each of the numbers from 1 to 2039 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)

2039 / 1 = 2039 (the remainder is 0, so 1 is a divisor of 2039)
2039 / 2 = 1019.5 (the remainder is 1, so 2 is not a divisor of 2039)
2039 / 3 = 679.66666666667 (the remainder is 2, so 3 is not a divisor of 2039)
...
2039 / 2038 = 1.0004906771344 (the remainder is 1, so 2038 is not a divisor of 2039)
2039 / 2039 = 1 (the remainder is 0, so 2039 is a divisor of 2039)

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 2039 (i.e. 45.155287619503). 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 $\frac{N}{d} = D$ )

2039 / 1 = 2039 (the remainder is 0, so 1 and 2039 are divisors of 2039)
2039 / 2 = 1019.5 (the remainder is 1, so 2 is not a divisor of 2039)
2039 / 3 = 679.66666666667 (the remainder is 2, so 3 is not a divisor of 2039)
...
2039 / 44 = 46.340909090909 (the remainder is 15, so 44 is not a divisor of 2039)
2039 / 45 = 45.311111111111 (the remainder is 14, so 45 is not a divisor of 2039)