What are the divisors of 5138?

1, 2, 7, 14, 367, 734, 2569, 5138

There is a total of 8 positive divisors.
The sum of these divisors is 8832.
The arithmetic mean is 1104.

4 even divisors

2, 14, 734, 5138

4 odd divisors

1, 7, 367, 2569

How to compute the divisors of 5138?

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 5138 by each of the numbers from 1 to 5138 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)

5138 / 1 = 5138 (the remainder is 0, so 1 is a divisor of 5138)
5138 / 2 = 2569 (the remainder is 0, so 2 is a divisor of 5138)
5138 / 3 = 1712.6666666667 (the remainder is 2, so 3 is not a divisor of 5138)
...
5138 / 5137 = 1.0001946661476 (the remainder is 1, so 5137 is not a divisor of 5138)
5138 / 5138 = 1 (the remainder is 0, so 5138 is a divisor of 5138)

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 5138 (i.e. 71.67984374983). 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 $\frac{N}{d} = D$ )

5138 / 1 = 5138 (the remainder is 0, so 1 and 5138 are divisors of 5138)
5138 / 2 = 2569 (the remainder is 0, so 2 and 2569 are divisors of 5138)
5138 / 3 = 1712.6666666667 (the remainder is 2, so 3 is not a divisor of 5138)
...
5138 / 70 = 73.4 (the remainder is 28, so 70 is not a divisor of 5138)
5138 / 71 = 72.366197183099 (the remainder is 26, so 71 is not a divisor of 5138)