What are the divisors of 4071?

1, 3, 23, 59, 69, 177, 1357, 4071

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, 23, 59, 69, 177, 1357, 4071

How to compute the divisors of 4071?

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 4071 by each of the numbers from 1 to 4071 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)

4071 / 1 = 4071 (the remainder is 0, so 1 is a divisor of 4071)
4071 / 2 = 2035.5 (the remainder is 1, so 2 is not a divisor of 4071)
4071 / 3 = 1357 (the remainder is 0, so 3 is a divisor of 4071)
...
4071 / 4070 = 1.0002457002457 (the remainder is 1, so 4070 is not a divisor of 4071)
4071 / 4071 = 1 (the remainder is 0, so 4071 is a divisor of 4071)

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 4071 (i.e. 63.804388563797). 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 $\frac{N}{d} = D$ )

4071 / 1 = 4071 (the remainder is 0, so 1 and 4071 are divisors of 4071)
4071 / 2 = 2035.5 (the remainder is 1, so 2 is not a divisor of 4071)
4071 / 3 = 1357 (the remainder is 0, so 3 and 1357 are divisors of 4071)
...
4071 / 62 = 65.661290322581 (the remainder is 41, so 62 is not a divisor of 4071)
4071 / 63 = 64.619047619048 (the remainder is 39, so 63 is not a divisor of 4071)