What are the divisors of 2138?

1, 2, 1069, 2138

There is a total of 4 positive divisors.
The sum of these divisors is 3210.
The arithmetic mean is 802.5.

2 even divisors

2, 2138

2 odd divisors

1, 1069

How to compute the divisors of 2138?

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 2138 by each of the numbers from 1 to 2138 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)

2138 / 1 = 2138 (the remainder is 0, so 1 is a divisor of 2138)
2138 / 2 = 1069 (the remainder is 0, so 2 is a divisor of 2138)
2138 / 3 = 712.66666666667 (the remainder is 2, so 3 is not a divisor of 2138)
...
2138 / 2137 = 1.0004679457183 (the remainder is 1, so 2137 is not a divisor of 2138)
2138 / 2138 = 1 (the remainder is 0, so 2138 is a divisor of 2138)

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 2138 (i.e. 46.238512086788). 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 $\frac{N}{d} = D$ )

2138 / 1 = 2138 (the remainder is 0, so 1 and 2138 are divisors of 2138)
2138 / 2 = 1069 (the remainder is 0, so 2 and 1069 are divisors of 2138)
2138 / 3 = 712.66666666667 (the remainder is 2, so 3 is not a divisor of 2138)
...
2138 / 45 = 47.511111111111 (the remainder is 23, so 45 is not a divisor of 2138)
2138 / 46 = 46.478260869565 (the remainder is 22, so 46 is not a divisor of 2138)