What are the divisors of 5731?

1, 11, 521, 5731

There is a total of 4 positive divisors.
The sum of these divisors is 6264.
The arithmetic mean is 1566.

4 odd divisors

1, 11, 521, 5731

How to compute the divisors of 5731?

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 5731 by each of the numbers from 1 to 5731 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)

5731 / 1 = 5731 (the remainder is 0, so 1 is a divisor of 5731)
5731 / 2 = 2865.5 (the remainder is 1, so 2 is not a divisor of 5731)
5731 / 3 = 1910.3333333333 (the remainder is 1, so 3 is not a divisor of 5731)
...
5731 / 5730 = 1.0001745200698 (the remainder is 1, so 5730 is not a divisor of 5731)
5731 / 5731 = 1 (the remainder is 0, so 5731 is a divisor of 5731)

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 5731 (i.e. 75.703368485161). 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 $\frac{N}{d} = D$ )

5731 / 1 = 5731 (the remainder is 0, so 1 and 5731 are divisors of 5731)
5731 / 2 = 2865.5 (the remainder is 1, so 2 is not a divisor of 5731)
5731 / 3 = 1910.3333333333 (the remainder is 1, so 3 is not a divisor of 5731)
...
5731 / 74 = 77.445945945946 (the remainder is 33, so 74 is not a divisor of 5731)
5731 / 75 = 76.413333333333 (the remainder is 31, so 75 is not a divisor of 5731)