What are the divisors of 5751?

1, 3, 9, 27, 71, 81, 213, 639, 1917, 5751

There is a total of 10 positive divisors.
The sum of these divisors is 8712.
The arithmetic mean is 871.2.

10 odd divisors

1, 3, 9, 27, 71, 81, 213, 639, 1917, 5751

How to compute the divisors of 5751?

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 5751 by each of the numbers from 1 to 5751 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)

5751 / 1 = 5751 (the remainder is 0, so 1 is a divisor of 5751)
5751 / 2 = 2875.5 (the remainder is 1, so 2 is not a divisor of 5751)
5751 / 3 = 1917 (the remainder is 0, so 3 is a divisor of 5751)
...
5751 / 5750 = 1.0001739130435 (the remainder is 1, so 5750 is not a divisor of 5751)
5751 / 5751 = 1 (the remainder is 0, so 5751 is a divisor of 5751)

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 5751 (i.e. 75.835347958587). 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 $\frac{N}{d} = D$ )

5751 / 1 = 5751 (the remainder is 0, so 1 and 5751 are divisors of 5751)
5751 / 2 = 2875.5 (the remainder is 1, so 2 is not a divisor of 5751)
5751 / 3 = 1917 (the remainder is 0, so 3 and 1917 are divisors of 5751)
...
5751 / 74 = 77.716216216216 (the remainder is 53, so 74 is not a divisor of 5751)
5751 / 75 = 76.68 (the remainder is 51, so 75 is not a divisor of 5751)