What are the divisors of 2019?

1, 3, 673, 2019

There is a total of 4 positive divisors.
The sum of these divisors is 2696.
The arithmetic mean is 674.

4 odd divisors

1, 3, 673, 2019

How to compute the divisors of 2019?

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 2019 by each of the numbers from 1 to 2019 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)

2019 / 1 = 2019 (the remainder is 0, so 1 is a divisor of 2019)
2019 / 2 = 1009.5 (the remainder is 1, so 2 is not a divisor of 2019)
2019 / 3 = 673 (the remainder is 0, so 3 is a divisor of 2019)
...
2019 / 2018 = 1.0004955401388 (the remainder is 1, so 2018 is not a divisor of 2019)
2019 / 2019 = 1 (the remainder is 0, so 2019 is a divisor of 2019)

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 2019 (i.e. 44.933283877322). 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 $\frac{N}{d} = D$ )

2019 / 1 = 2019 (the remainder is 0, so 1 and 2019 are divisors of 2019)
2019 / 2 = 1009.5 (the remainder is 1, so 2 is not a divisor of 2019)
2019 / 3 = 673 (the remainder is 0, so 3 and 673 are divisors of 2019)
...
2019 / 43 = 46.953488372093 (the remainder is 41, so 43 is not a divisor of 2019)
2019 / 44 = 45.886363636364 (the remainder is 39, so 44 is not a divisor of 2019)