What are the divisors of 4085?

1, 5, 19, 43, 95, 215, 817, 4085

There is a total of 8 positive divisors.
The sum of these divisors is 5280.
The arithmetic mean is 660.

8 odd divisors

1, 5, 19, 43, 95, 215, 817, 4085

How to compute the divisors of 4085?

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 4085 by each of the numbers from 1 to 4085 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)

4085 / 1 = 4085 (the remainder is 0, so 1 is a divisor of 4085)
4085 / 2 = 2042.5 (the remainder is 1, so 2 is not a divisor of 4085)
4085 / 3 = 1361.6666666667 (the remainder is 2, so 3 is not a divisor of 4085)
...
4085 / 4084 = 1.0002448579824 (the remainder is 1, so 4084 is not a divisor of 4085)
4085 / 4085 = 1 (the remainder is 0, so 4085 is a divisor of 4085)

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 4085 (i.e. 63.914004725099). 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 $\frac{N}{d} = D$ )

4085 / 1 = 4085 (the remainder is 0, so 1 and 4085 are divisors of 4085)
4085 / 2 = 2042.5 (the remainder is 1, so 2 is not a divisor of 4085)
4085 / 3 = 1361.6666666667 (the remainder is 2, so 3 is not a divisor of 4085)
...
4085 / 62 = 65.887096774194 (the remainder is 55, so 62 is not a divisor of 4085)
4085 / 63 = 64.84126984127 (the remainder is 53, so 63 is not a divisor of 4085)