What are the divisors of 4115?

1, 5, 823, 4115

There is a total of 4 positive divisors.
The sum of these divisors is 4944.
The arithmetic mean is 1236.

4 odd divisors

1, 5, 823, 4115

How to compute the divisors of 4115?

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 4115 by each of the numbers from 1 to 4115 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)

4115 / 1 = 4115 (the remainder is 0, so 1 is a divisor of 4115)
4115 / 2 = 2057.5 (the remainder is 1, so 2 is not a divisor of 4115)
4115 / 3 = 1371.6666666667 (the remainder is 2, so 3 is not a divisor of 4115)
...
4115 / 4114 = 1.0002430724356 (the remainder is 1, so 4114 is not a divisor of 4115)
4115 / 4115 = 1 (the remainder is 0, so 4115 is a divisor of 4115)

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 4115 (i.e. 64.148265759878). 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 $\frac{N}{d} = D$ )

4115 / 1 = 4115 (the remainder is 0, so 1 and 4115 are divisors of 4115)
4115 / 2 = 2057.5 (the remainder is 1, so 2 is not a divisor of 4115)
4115 / 3 = 1371.6666666667 (the remainder is 2, so 3 is not a divisor of 4115)
...
4115 / 63 = 65.31746031746 (the remainder is 20, so 63 is not a divisor of 4115)
4115 / 64 = 64.296875 (the remainder is 19, so 64 is not a divisor of 4115)