What are the divisors of 4089?

1, 3, 29, 47, 87, 141, 1363, 4089

There is a total of 8 positive divisors.
The sum of these divisors is 5760.
The arithmetic mean is 720.

8 odd divisors

1, 3, 29, 47, 87, 141, 1363, 4089

How to compute the divisors of 4089?

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 4089 by each of the numbers from 1 to 4089 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)

4089 / 1 = 4089 (the remainder is 0, so 1 is a divisor of 4089)
4089 / 2 = 2044.5 (the remainder is 1, so 2 is not a divisor of 4089)
4089 / 3 = 1363 (the remainder is 0, so 3 is a divisor of 4089)
...
4089 / 4088 = 1.0002446183953 (the remainder is 1, so 4088 is not a divisor of 4089)
4089 / 4089 = 1 (the remainder is 0, so 4089 is a divisor of 4089)

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 4089 (i.e. 63.945289114993). 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 $\frac{N}{d} = D$ )

4089 / 1 = 4089 (the remainder is 0, so 1 and 4089 are divisors of 4089)
4089 / 2 = 2044.5 (the remainder is 1, so 2 is not a divisor of 4089)
4089 / 3 = 1363 (the remainder is 0, so 3 and 1363 are divisors of 4089)
...
4089 / 62 = 65.951612903226 (the remainder is 59, so 62 is not a divisor of 4089)
4089 / 63 = 64.904761904762 (the remainder is 57, so 63 is not a divisor of 4089)