What are the divisors of 680?

1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680

There is a total of 16 positive divisors.
The sum of these divisors is 1620.
The arithmetic mean is 101.25.

12 even divisors

2, 4, 8, 10, 20, 34, 40, 68, 136, 170, 340, 680

4 odd divisors

1, 5, 17, 85

How to compute the divisors of 680?

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 680 by each of the numbers from 1 to 680 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)

680 / 1 = 680 (the remainder is 0, so 1 is a divisor of 680)
680 / 2 = 340 (the remainder is 0, so 2 is a divisor of 680)
680 / 3 = 226.66666666667 (the remainder is 2, so 3 is not a divisor of 680)
...
680 / 679 = 1.0014727540501 (the remainder is 1, so 679 is not a divisor of 680)
680 / 680 = 1 (the remainder is 0, so 680 is a divisor of 680)

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 680 (i.e. 26.076809620811). 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 $\frac{N}{d} = D$ )

680 / 1 = 680 (the remainder is 0, so 1 and 680 are divisors of 680)
680 / 2 = 340 (the remainder is 0, so 2 and 340 are divisors of 680)
680 / 3 = 226.66666666667 (the remainder is 2, so 3 is not a divisor of 680)
...
680 / 25 = 27.2 (the remainder is 5, so 25 is not a divisor of 680)
680 / 26 = 26.153846153846 (the remainder is 4, so 26 is not a divisor of 680)