What are the divisors of 1684?

1, 2, 4, 421, 842, 1684

There is a total of 6 positive divisors.
The sum of these divisors is 2954.
The arithmetic mean is 492.33333333333.

4 even divisors

2, 4, 842, 1684

2 odd divisors

1, 421

How to compute the divisors of 1684?

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 1684 by each of the numbers from 1 to 1684 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)

1684 / 1 = 1684 (the remainder is 0, so 1 is a divisor of 1684)
1684 / 2 = 842 (the remainder is 0, so 2 is a divisor of 1684)
1684 / 3 = 561.33333333333 (the remainder is 1, so 3 is not a divisor of 1684)
...
1684 / 1683 = 1.0005941770648 (the remainder is 1, so 1683 is not a divisor of 1684)
1684 / 1684 = 1 (the remainder is 0, so 1684 is a divisor of 1684)

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 1684 (i.e. 41.036569057366). 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 $\frac{N}{d} = D$ )

1684 / 1 = 1684 (the remainder is 0, so 1 and 1684 are divisors of 1684)
1684 / 2 = 842 (the remainder is 0, so 2 and 842 are divisors of 1684)
1684 / 3 = 561.33333333333 (the remainder is 1, so 3 is not a divisor of 1684)
...
1684 / 40 = 42.1 (the remainder is 4, so 40 is not a divisor of 1684)
1684 / 41 = 41.073170731707 (the remainder is 3, so 41 is not a divisor of 1684)