What are the divisors of 688?

1, 2, 4, 8, 16, 43, 86, 172, 344, 688

There is a total of 10 positive divisors.
The sum of these divisors is 1364.
The arithmetic mean is 136.4.

8 even divisors

2, 4, 8, 16, 86, 172, 344, 688

2 odd divisors

1, 43

How to compute the divisors of 688?

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 688 by each of the numbers from 1 to 688 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)

688 / 1 = 688 (the remainder is 0, so 1 is a divisor of 688)
688 / 2 = 344 (the remainder is 0, so 2 is a divisor of 688)
688 / 3 = 229.33333333333 (the remainder is 1, so 3 is not a divisor of 688)
...
688 / 687 = 1.0014556040757 (the remainder is 1, so 687 is not a divisor of 688)
688 / 688 = 1 (the remainder is 0, so 688 is a divisor of 688)

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 688 (i.e. 26.229754097208). 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 $\frac{N}{d} = D$ )

688 / 1 = 688 (the remainder is 0, so 1 and 688 are divisors of 688)
688 / 2 = 344 (the remainder is 0, so 2 and 344 are divisors of 688)
688 / 3 = 229.33333333333 (the remainder is 1, so 3 is not a divisor of 688)
...
688 / 25 = 27.52 (the remainder is 13, so 25 is not a divisor of 688)
688 / 26 = 26.461538461538 (the remainder is 12, so 26 is not a divisor of 688)