What are the divisors of 4165?

1, 5, 7, 17, 35, 49, 85, 119, 245, 595, 833, 4165

There is a total of 12 positive divisors.
The sum of these divisors is 6156.
The arithmetic mean is 513.

12 odd divisors

1, 5, 7, 17, 35, 49, 85, 119, 245, 595, 833, 4165

How to compute the divisors of 4165?

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 4165 by each of the numbers from 1 to 4165 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)

4165 / 1 = 4165 (the remainder is 0, so 1 is a divisor of 4165)
4165 / 2 = 2082.5 (the remainder is 1, so 2 is not a divisor of 4165)
4165 / 3 = 1388.3333333333 (the remainder is 1, so 3 is not a divisor of 4165)
...
4165 / 4164 = 1.0002401536984 (the remainder is 1, so 4164 is not a divisor of 4165)
4165 / 4165 = 1 (the remainder is 0, so 4165 is a divisor of 4165)

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 4165 (i.e. 64.53681120105). 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 $\frac{N}{d} = D$ )

4165 / 1 = 4165 (the remainder is 0, so 1 and 4165 are divisors of 4165)
4165 / 2 = 2082.5 (the remainder is 1, so 2 is not a divisor of 4165)
4165 / 3 = 1388.3333333333 (the remainder is 1, so 3 is not a divisor of 4165)
...
4165 / 63 = 66.111111111111 (the remainder is 7, so 63 is not a divisor of 4165)
4165 / 64 = 65.078125 (the remainder is 5, so 64 is not a divisor of 4165)