What are the divisors of 4109?

1, 7, 587, 4109

There is a total of 4 positive divisors.
The sum of these divisors is 4704.
The arithmetic mean is 1176.

4 odd divisors

1, 7, 587, 4109

How to compute the divisors of 4109?

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 4109 by each of the numbers from 1 to 4109 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)

4109 / 1 = 4109 (the remainder is 0, so 1 is a divisor of 4109)
4109 / 2 = 2054.5 (the remainder is 1, so 2 is not a divisor of 4109)
4109 / 3 = 1369.6666666667 (the remainder is 2, so 3 is not a divisor of 4109)
...
4109 / 4108 = 1.0002434274586 (the remainder is 1, so 4108 is not a divisor of 4109)
4109 / 4109 = 1 (the remainder is 0, so 4109 is a divisor of 4109)

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 4109 (i.e. 64.101482042149). 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 $\frac{N}{d} = D$ )

4109 / 1 = 4109 (the remainder is 0, so 1 and 4109 are divisors of 4109)
4109 / 2 = 2054.5 (the remainder is 1, so 2 is not a divisor of 4109)
4109 / 3 = 1369.6666666667 (the remainder is 2, so 3 is not a divisor of 4109)
...
4109 / 63 = 65.222222222222 (the remainder is 14, so 63 is not a divisor of 4109)
4109 / 64 = 64.203125 (the remainder is 13, so 64 is not a divisor of 4109)