What are the divisors of 2139?

1, 3, 23, 31, 69, 93, 713, 2139

There is a total of 8 positive divisors.
The sum of these divisors is 3072.
The arithmetic mean is 384.

8 odd divisors

1, 3, 23, 31, 69, 93, 713, 2139

How to compute the divisors of 2139?

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 2139 by each of the numbers from 1 to 2139 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)

2139 / 1 = 2139 (the remainder is 0, so 1 is a divisor of 2139)
2139 / 2 = 1069.5 (the remainder is 1, so 2 is not a divisor of 2139)
2139 / 3 = 713 (the remainder is 0, so 3 is a divisor of 2139)
...
2139 / 2138 = 1.0004677268475 (the remainder is 1, so 2138 is not a divisor of 2139)
2139 / 2139 = 1 (the remainder is 0, so 2139 is a divisor of 2139)

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 2139 (i.e. 46.249324319389). 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 $\frac{N}{d} = D$ )

2139 / 1 = 2139 (the remainder is 0, so 1 and 2139 are divisors of 2139)
2139 / 2 = 1069.5 (the remainder is 1, so 2 is not a divisor of 2139)
2139 / 3 = 713 (the remainder is 0, so 3 and 713 are divisors of 2139)
...
2139 / 45 = 47.533333333333 (the remainder is 24, so 45 is not a divisor of 2139)
2139 / 46 = 46.5 (the remainder is 23, so 46 is not a divisor of 2139)