What are the divisors of 1689?

1, 3, 563, 1689

There is a total of 4 positive divisors.
The sum of these divisors is 2256.
The arithmetic mean is 564.

4 odd divisors

1, 3, 563, 1689

How to compute the divisors of 1689?

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 1689 by each of the numbers from 1 to 1689 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)

1689 / 1 = 1689 (the remainder is 0, so 1 is a divisor of 1689)
1689 / 2 = 844.5 (the remainder is 1, so 2 is not a divisor of 1689)
1689 / 3 = 563 (the remainder is 0, so 3 is a divisor of 1689)
...
1689 / 1688 = 1.0005924170616 (the remainder is 1, so 1688 is not a divisor of 1689)
1689 / 1689 = 1 (the remainder is 0, so 1689 is a divisor of 1689)

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 1689 (i.e. 41.097445176069). 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 $\frac{N}{d} = D$ )

1689 / 1 = 1689 (the remainder is 0, so 1 and 1689 are divisors of 1689)
1689 / 2 = 844.5 (the remainder is 1, so 2 is not a divisor of 1689)
1689 / 3 = 563 (the remainder is 0, so 3 and 563 are divisors of 1689)
...
1689 / 40 = 42.225 (the remainder is 9, so 40 is not a divisor of 1689)
1689 / 41 = 41.19512195122 (the remainder is 8, so 41 is not a divisor of 1689)