What are the divisors of 686?

1, 2, 7, 14, 49, 98, 343, 686

There is a total of 8 positive divisors.
The sum of these divisors is 1200.
The arithmetic mean is 150.

4 even divisors

2, 14, 98, 686

4 odd divisors

1, 7, 49, 343

How to compute the divisors of 686?

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 686 by each of the numbers from 1 to 686 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)

686 / 1 = 686 (the remainder is 0, so 1 is a divisor of 686)
686 / 2 = 343 (the remainder is 0, so 2 is a divisor of 686)
686 / 3 = 228.66666666667 (the remainder is 2, so 3 is not a divisor of 686)
...
686 / 685 = 1.0014598540146 (the remainder is 1, so 685 is not a divisor of 686)
686 / 686 = 1 (the remainder is 0, so 686 is a divisor of 686)

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 686 (i.e. 26.191601707418). 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 $\frac{N}{d} = D$ )

686 / 1 = 686 (the remainder is 0, so 1 and 686 are divisors of 686)
686 / 2 = 343 (the remainder is 0, so 2 and 343 are divisors of 686)
686 / 3 = 228.66666666667 (the remainder is 2, so 3 is not a divisor of 686)
...
686 / 25 = 27.44 (the remainder is 11, so 25 is not a divisor of 686)
686 / 26 = 26.384615384615 (the remainder is 10, so 26 is not a divisor of 686)