What are the divisors of 4113?

1, 3, 9, 457, 1371, 4113

There is a total of 6 positive divisors.
The sum of these divisors is 5954.
The arithmetic mean is 992.33333333333.

6 odd divisors

1, 3, 9, 457, 1371, 4113

How to compute the divisors of 4113?

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 4113 by each of the numbers from 1 to 4113 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)

4113 / 1 = 4113 (the remainder is 0, so 1 is a divisor of 4113)
4113 / 2 = 2056.5 (the remainder is 1, so 2 is not a divisor of 4113)
4113 / 3 = 1371 (the remainder is 0, so 3 is a divisor of 4113)
...
4113 / 4112 = 1.0002431906615 (the remainder is 1, so 4112 is not a divisor of 4113)
4113 / 4113 = 1 (the remainder is 0, so 4113 is a divisor of 4113)

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 4113 (i.e. 64.132674979296). 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 $\frac{N}{d} = D$ )

4113 / 1 = 4113 (the remainder is 0, so 1 and 4113 are divisors of 4113)
4113 / 2 = 2056.5 (the remainder is 1, so 2 is not a divisor of 4113)
4113 / 3 = 1371 (the remainder is 0, so 3 and 1371 are divisors of 4113)
...
4113 / 63 = 65.285714285714 (the remainder is 18, so 63 is not a divisor of 4113)
4113 / 64 = 64.265625 (the remainder is 17, so 64 is not a divisor of 4113)