What are the divisors of 689?

1, 13, 53, 689

There is a total of 4 positive divisors.
The sum of these divisors is 756.
The arithmetic mean is 189.

4 odd divisors

1, 13, 53, 689

How to compute the divisors of 689?

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 689 by each of the numbers from 1 to 689 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)

689 / 1 = 689 (the remainder is 0, so 1 is a divisor of 689)
689 / 2 = 344.5 (the remainder is 1, so 2 is not a divisor of 689)
689 / 3 = 229.66666666667 (the remainder is 2, so 3 is not a divisor of 689)
...
689 / 688 = 1.0014534883721 (the remainder is 1, so 688 is not a divisor of 689)
689 / 689 = 1 (the remainder is 0, so 689 is a divisor of 689)

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 689 (i.e. 26.248809496813). 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 $\frac{N}{d} = D$ )

689 / 1 = 689 (the remainder is 0, so 1 and 689 are divisors of 689)
689 / 2 = 344.5 (the remainder is 1, so 2 is not a divisor of 689)
689 / 3 = 229.66666666667 (the remainder is 2, so 3 is not a divisor of 689)
...
689 / 25 = 27.56 (the remainder is 14, so 25 is not a divisor of 689)
689 / 26 = 26.5 (the remainder is 13, so 26 is not a divisor of 689)