What are the divisors of 1686?

1, 2, 3, 6, 281, 562, 843, 1686

There is a total of 8 positive divisors.
The sum of these divisors is 3384.
The arithmetic mean is 423.

4 even divisors

2, 6, 562, 1686

4 odd divisors

1, 3, 281, 843

How to compute the divisors of 1686?

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 1686 by each of the numbers from 1 to 1686 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)

1686 / 1 = 1686 (the remainder is 0, so 1 is a divisor of 1686)
1686 / 2 = 843 (the remainder is 0, so 2 is a divisor of 1686)
1686 / 3 = 562 (the remainder is 0, so 3 is a divisor of 1686)
...
1686 / 1685 = 1.0005934718101 (the remainder is 1, so 1685 is not a divisor of 1686)
1686 / 1686 = 1 (the remainder is 0, so 1686 is a divisor of 1686)

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 1686 (i.e. 41.060930335296). 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 $\frac{N}{d} = D$ )

1686 / 1 = 1686 (the remainder is 0, so 1 and 1686 are divisors of 1686)
1686 / 2 = 843 (the remainder is 0, so 2 and 843 are divisors of 1686)
1686 / 3 = 562 (the remainder is 0, so 3 and 562 are divisors of 1686)
...
1686 / 40 = 42.15 (the remainder is 6, so 40 is not a divisor of 1686)
1686 / 41 = 41.121951219512 (the remainder is 5, so 41 is not a divisor of 1686)