What are the divisors of 714?

1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714

There is a total of 16 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 108.

8 even divisors

2, 6, 14, 34, 42, 102, 238, 714

8 odd divisors

1, 3, 7, 17, 21, 51, 119, 357

How to compute the divisors of 714?

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 714 by each of the numbers from 1 to 714 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)

714 / 1 = 714 (the remainder is 0, so 1 is a divisor of 714)
714 / 2 = 357 (the remainder is 0, so 2 is a divisor of 714)
714 / 3 = 238 (the remainder is 0, so 3 is a divisor of 714)
...
714 / 713 = 1.0014025245442 (the remainder is 1, so 713 is not a divisor of 714)
714 / 714 = 1 (the remainder is 0, so 714 is a divisor of 714)

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 714 (i.e. 26.720778431775). 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 $\frac{N}{d} = D$ )

714 / 1 = 714 (the remainder is 0, so 1 and 714 are divisors of 714)
714 / 2 = 357 (the remainder is 0, so 2 and 357 are divisors of 714)
714 / 3 = 238 (the remainder is 0, so 3 and 238 are divisors of 714)
...
714 / 25 = 28.56 (the remainder is 14, so 25 is not a divisor of 714)
714 / 26 = 27.461538461538 (the remainder is 12, so 26 is not a divisor of 714)