What are the divisors of 717?

1, 3, 239, 717

There is a total of 4 positive divisors.
The sum of these divisors is 960.
The arithmetic mean is 240.

4 odd divisors

1, 3, 239, 717

How to compute the divisors of 717?

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 717 by each of the numbers from 1 to 717 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)

717 / 1 = 717 (the remainder is 0, so 1 is a divisor of 717)
717 / 2 = 358.5 (the remainder is 1, so 2 is not a divisor of 717)
717 / 3 = 239 (the remainder is 0, so 3 is a divisor of 717)
...
717 / 716 = 1.0013966480447 (the remainder is 1, so 716 is not a divisor of 717)
717 / 717 = 1 (the remainder is 0, so 717 is a divisor of 717)

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 717 (i.e. 26.776855677992). 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 $\frac{N}{d} = D$ )

717 / 1 = 717 (the remainder is 0, so 1 and 717 are divisors of 717)
717 / 2 = 358.5 (the remainder is 1, so 2 is not a divisor of 717)
717 / 3 = 239 (the remainder is 0, so 3 and 239 are divisors of 717)
...
717 / 25 = 28.68 (the remainder is 17, so 25 is not a divisor of 717)
717 / 26 = 27.576923076923 (the remainder is 15, so 26 is not a divisor of 717)