What are the divisors of 682?

1, 2, 11, 22, 31, 62, 341, 682

There is a total of 8 positive divisors.
The sum of these divisors is 1152.
The arithmetic mean is 144.

4 even divisors

2, 22, 62, 682

4 odd divisors

1, 11, 31, 341

How to compute the divisors of 682?

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 682 by each of the numbers from 1 to 682 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)

682 / 1 = 682 (the remainder is 0, so 1 is a divisor of 682)
682 / 2 = 341 (the remainder is 0, so 2 is a divisor of 682)
682 / 3 = 227.33333333333 (the remainder is 1, so 3 is not a divisor of 682)
...
682 / 681 = 1.0014684287812 (the remainder is 1, so 681 is not a divisor of 682)
682 / 682 = 1 (the remainder is 0, so 682 is a divisor of 682)

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 682 (i.e. 26.115129714401). 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 $\frac{N}{d} = D$ )

682 / 1 = 682 (the remainder is 0, so 1 and 682 are divisors of 682)
682 / 2 = 341 (the remainder is 0, so 2 and 341 are divisors of 682)
682 / 3 = 227.33333333333 (the remainder is 1, so 3 is not a divisor of 682)
...
682 / 25 = 27.28 (the remainder is 7, so 25 is not a divisor of 682)
682 / 26 = 26.230769230769 (the remainder is 6, so 26 is not a divisor of 682)