What are the divisors of 4591?

1, 4591

There is a total of 2 positive divisors.
The sum of these divisors is 4592.
The arithmetic mean is 2296.

2 odd divisors

1, 4591

How to compute the divisors of 4591?

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 4591 by each of the numbers from 1 to 4591 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)

4591 / 1 = 4591 (the remainder is 0, so 1 is a divisor of 4591)
4591 / 2 = 2295.5 (the remainder is 1, so 2 is not a divisor of 4591)
4591 / 3 = 1530.3333333333 (the remainder is 1, so 3 is not a divisor of 4591)
...
4591 / 4590 = 1.0002178649237 (the remainder is 1, so 4590 is not a divisor of 4591)
4591 / 4591 = 1 (the remainder is 0, so 4591 is a divisor of 4591)

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 4591 (i.e. 67.75691846594). 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 $\frac{N}{d} = D$ )

4591 / 1 = 4591 (the remainder is 0, so 1 and 4591 are divisors of 4591)
4591 / 2 = 2295.5 (the remainder is 1, so 2 is not a divisor of 4591)
4591 / 3 = 1530.3333333333 (the remainder is 1, so 3 is not a divisor of 4591)
...
4591 / 66 = 69.560606060606 (the remainder is 37, so 66 is not a divisor of 4591)
4591 / 67 = 68.522388059701 (the remainder is 35, so 67 is not a divisor of 4591)