What are the divisors of 4161?

1, 3, 19, 57, 73, 219, 1387, 4161

There is a total of 8 positive divisors.
The sum of these divisors is 5920.
The arithmetic mean is 740.

8 odd divisors

1, 3, 19, 57, 73, 219, 1387, 4161

How to compute the divisors of 4161?

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 4161 by each of the numbers from 1 to 4161 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)

4161 / 1 = 4161 (the remainder is 0, so 1 is a divisor of 4161)
4161 / 2 = 2080.5 (the remainder is 1, so 2 is not a divisor of 4161)
4161 / 3 = 1387 (the remainder is 0, so 3 is a divisor of 4161)
...
4161 / 4160 = 1.0002403846154 (the remainder is 1, so 4160 is not a divisor of 4161)
4161 / 4161 = 1 (the remainder is 0, so 4161 is a divisor of 4161)

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 4161 (i.e. 64.505813691481). 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 $\frac{N}{d} = D$ )

4161 / 1 = 4161 (the remainder is 0, so 1 and 4161 are divisors of 4161)
4161 / 2 = 2080.5 (the remainder is 1, so 2 is not a divisor of 4161)
4161 / 3 = 1387 (the remainder is 0, so 3 and 1387 are divisors of 4161)
...
4161 / 63 = 66.047619047619 (the remainder is 3, so 63 is not a divisor of 4161)
4161 / 64 = 65.015625 (the remainder is 1, so 64 is not a divisor of 4161)