What are the divisors of 4131?

1, 3, 9, 17, 27, 51, 81, 153, 243, 459, 1377, 4131

There is a total of 12 positive divisors.
The sum of these divisors is 6552.
The arithmetic mean is 546.

12 odd divisors

1, 3, 9, 17, 27, 51, 81, 153, 243, 459, 1377, 4131

How to compute the divisors of 4131?

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 4131 by each of the numbers from 1 to 4131 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)

4131 / 1 = 4131 (the remainder is 0, so 1 is a divisor of 4131)
4131 / 2 = 2065.5 (the remainder is 1, so 2 is not a divisor of 4131)
4131 / 3 = 1377 (the remainder is 0, so 3 is a divisor of 4131)
...
4131 / 4130 = 1.0002421307506 (the remainder is 1, so 4130 is not a divisor of 4131)
4131 / 4131 = 1 (the remainder is 0, so 4131 is a divisor of 4131)

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 4131 (i.e. 64.272855856886). 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 $\frac{N}{d} = D$ )

4131 / 1 = 4131 (the remainder is 0, so 1 and 4131 are divisors of 4131)
4131 / 2 = 2065.5 (the remainder is 1, so 2 is not a divisor of 4131)
4131 / 3 = 1377 (the remainder is 0, so 3 and 1377 are divisors of 4131)
...
4131 / 63 = 65.571428571429 (the remainder is 36, so 63 is not a divisor of 4131)
4131 / 64 = 64.546875 (the remainder is 35, so 64 is not a divisor of 4131)