What are the divisors of 2067?

1, 3, 13, 39, 53, 159, 689, 2067

There is a total of 8 positive divisors.
The sum of these divisors is 3024.
The arithmetic mean is 378.

8 odd divisors

1, 3, 13, 39, 53, 159, 689, 2067

How to compute the divisors of 2067?

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 2067 by each of the numbers from 1 to 2067 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)

2067 / 1 = 2067 (the remainder is 0, so 1 is a divisor of 2067)
2067 / 2 = 1033.5 (the remainder is 1, so 2 is not a divisor of 2067)
2067 / 3 = 689 (the remainder is 0, so 3 is a divisor of 2067)
...
2067 / 2066 = 1.0004840271055 (the remainder is 1, so 2066 is not a divisor of 2067)
2067 / 2067 = 1 (the remainder is 0, so 2067 is a divisor of 2067)

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 2067 (i.e. 45.464271686677). 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 $\frac{N}{d} = D$ )

2067 / 1 = 2067 (the remainder is 0, so 1 and 2067 are divisors of 2067)
2067 / 2 = 1033.5 (the remainder is 1, so 2 is not a divisor of 2067)
2067 / 3 = 689 (the remainder is 0, so 3 and 689 are divisors of 2067)
...
2067 / 44 = 46.977272727273 (the remainder is 43, so 44 is not a divisor of 2067)
2067 / 45 = 45.933333333333 (the remainder is 42, so 45 is not a divisor of 2067)