What are the divisors of 2017?

1, 2017

There is a total of 2 positive divisors.
The sum of these divisors is 2018.
The arithmetic mean is 1009.

2 odd divisors

1, 2017

How to compute the divisors of 2017?

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 2017 by each of the numbers from 1 to 2017 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)

2017 / 1 = 2017 (the remainder is 0, so 1 is a divisor of 2017)
2017 / 2 = 1008.5 (the remainder is 1, so 2 is not a divisor of 2017)
2017 / 3 = 672.33333333333 (the remainder is 1, so 3 is not a divisor of 2017)
...
2017 / 2016 = 1.000496031746 (the remainder is 1, so 2016 is not a divisor of 2017)
2017 / 2017 = 1 (the remainder is 0, so 2017 is a divisor of 2017)

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 2017 (i.e. 44.911023145771). 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 $\frac{N}{d} = D$ )

2017 / 1 = 2017 (the remainder is 0, so 1 and 2017 are divisors of 2017)
2017 / 2 = 1008.5 (the remainder is 1, so 2 is not a divisor of 2017)
2017 / 3 = 672.33333333333 (the remainder is 1, so 3 is not a divisor of 2017)
...
2017 / 43 = 46.906976744186 (the remainder is 39, so 43 is not a divisor of 2017)
2017 / 44 = 45.840909090909 (the remainder is 37, so 44 is not a divisor of 2017)