What are the divisors of 4139?

1, 4139

There is a total of 2 positive divisors.
The sum of these divisors is 4140.
The arithmetic mean is 2070.

2 odd divisors

1, 4139

How to compute the divisors of 4139?

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 4139 by each of the numbers from 1 to 4139 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)

4139 / 1 = 4139 (the remainder is 0, so 1 is a divisor of 4139)
4139 / 2 = 2069.5 (the remainder is 1, so 2 is not a divisor of 4139)
4139 / 3 = 1379.6666666667 (the remainder is 2, so 3 is not a divisor of 4139)
...
4139 / 4138 = 1.000241662639 (the remainder is 1, so 4138 is not a divisor of 4139)
4139 / 4139 = 1 (the remainder is 0, so 4139 is a divisor of 4139)

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 4139 (i.e. 64.335060425867). 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 $\frac{N}{d} = D$ )

4139 / 1 = 4139 (the remainder is 0, so 1 and 4139 are divisors of 4139)
4139 / 2 = 2069.5 (the remainder is 1, so 2 is not a divisor of 4139)
4139 / 3 = 1379.6666666667 (the remainder is 2, so 3 is not a divisor of 4139)
...
4139 / 63 = 65.698412698413 (the remainder is 44, so 63 is not a divisor of 4139)
4139 / 64 = 64.671875 (the remainder is 43, so 64 is not a divisor of 4139)