What are the divisors of 4138?

1, 2, 2069, 4138

There is a total of 4 positive divisors.
The sum of these divisors is 6210.
The arithmetic mean is 1552.5.

2 even divisors

2, 4138

2 odd divisors

1, 2069

How to compute the divisors of 4138?

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 4138 by each of the numbers from 1 to 4138 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)

4138 / 1 = 4138 (the remainder is 0, so 1 is a divisor of 4138)
4138 / 2 = 2069 (the remainder is 0, so 2 is a divisor of 4138)
4138 / 3 = 1379.3333333333 (the remainder is 1, so 3 is not a divisor of 4138)
...
4138 / 4137 = 1.0002417210539 (the remainder is 1, so 4137 is not a divisor of 4138)
4138 / 4138 = 1 (the remainder is 0, so 4138 is a divisor of 4138)

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 4138 (i.e. 64.327288144302). 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 $\frac{N}{d} = D$ )

4138 / 1 = 4138 (the remainder is 0, so 1 and 4138 are divisors of 4138)
4138 / 2 = 2069 (the remainder is 0, so 2 and 2069 are divisors of 4138)
4138 / 3 = 1379.3333333333 (the remainder is 1, so 3 is not a divisor of 4138)
...
4138 / 63 = 65.68253968254 (the remainder is 43, so 63 is not a divisor of 4138)
4138 / 64 = 64.65625 (the remainder is 42, so 64 is not a divisor of 4138)