What are the divisors of 2031?

1, 3, 677, 2031

There is a total of 4 positive divisors.
The sum of these divisors is 2712.
The arithmetic mean is 678.

4 odd divisors

1, 3, 677, 2031

How to compute the divisors of 2031?

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 2031 by each of the numbers from 1 to 2031 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)

2031 / 1 = 2031 (the remainder is 0, so 1 is a divisor of 2031)
2031 / 2 = 1015.5 (the remainder is 1, so 2 is not a divisor of 2031)
2031 / 3 = 677 (the remainder is 0, so 3 is a divisor of 2031)
...
2031 / 2030 = 1.0004926108374 (the remainder is 1, so 2030 is not a divisor of 2031)
2031 / 2031 = 1 (the remainder is 0, so 2031 is a divisor of 2031)

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 2031 (i.e. 45.066617356975). 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 $\frac{N}{d} = D$ )

2031 / 1 = 2031 (the remainder is 0, so 1 and 2031 are divisors of 2031)
2031 / 2 = 1015.5 (the remainder is 1, so 2 is not a divisor of 2031)
2031 / 3 = 677 (the remainder is 0, so 3 and 677 are divisors of 2031)
...
2031 / 44 = 46.159090909091 (the remainder is 7, so 44 is not a divisor of 2031)
2031 / 45 = 45.133333333333 (the remainder is 6, so 45 is not a divisor of 2031)