What are the divisors of 711?

1, 3, 9, 79, 237, 711

There is a total of 6 positive divisors.
The sum of these divisors is 1040.
The arithmetic mean is 173.33333333333.

6 odd divisors

1, 3, 9, 79, 237, 711

How to compute the divisors of 711?

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 711 by each of the numbers from 1 to 711 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)

711 / 1 = 711 (the remainder is 0, so 1 is a divisor of 711)
711 / 2 = 355.5 (the remainder is 1, so 2 is not a divisor of 711)
711 / 3 = 237 (the remainder is 0, so 3 is a divisor of 711)
...
711 / 710 = 1.0014084507042 (the remainder is 1, so 710 is not a divisor of 711)
711 / 711 = 1 (the remainder is 0, so 711 is a divisor of 711)

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 711 (i.e. 26.664583251947). 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 $\frac{N}{d} = D$ )

711 / 1 = 711 (the remainder is 0, so 1 and 711 are divisors of 711)
711 / 2 = 355.5 (the remainder is 1, so 2 is not a divisor of 711)
711 / 3 = 237 (the remainder is 0, so 3 and 237 are divisors of 711)
...
711 / 25 = 28.44 (the remainder is 11, so 25 is not a divisor of 711)
711 / 26 = 27.346153846154 (the remainder is 9, so 26 is not a divisor of 711)