What are the divisors of 4091?

1, 4091

There is a total of 2 positive divisors.
The sum of these divisors is 4092.
The arithmetic mean is 2046.

2 odd divisors

1, 4091

How to compute the divisors of 4091?

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 4091 by each of the numbers from 1 to 4091 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)

4091 / 1 = 4091 (the remainder is 0, so 1 is a divisor of 4091)
4091 / 2 = 2045.5 (the remainder is 1, so 2 is not a divisor of 4091)
4091 / 3 = 1363.6666666667 (the remainder is 2, so 3 is not a divisor of 4091)
...
4091 / 4090 = 1.0002444987775 (the remainder is 1, so 4090 is not a divisor of 4091)
4091 / 4091 = 1 (the remainder is 0, so 4091 is a divisor of 4091)

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 4091 (i.e. 63.96092557179). 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 $\frac{N}{d} = D$ )

4091 / 1 = 4091 (the remainder is 0, so 1 and 4091 are divisors of 4091)
4091 / 2 = 2045.5 (the remainder is 1, so 2 is not a divisor of 4091)
4091 / 3 = 1363.6666666667 (the remainder is 2, so 3 is not a divisor of 4091)
...
4091 / 62 = 65.983870967742 (the remainder is 61, so 62 is not a divisor of 4091)
4091 / 63 = 64.936507936508 (the remainder is 59, so 63 is not a divisor of 4091)