What are the divisors of 1680?

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680

There is a total of 40 positive divisors.
The sum of these divisors is 5952.
The arithmetic mean is 148.8.

32 even divisors

2, 4, 6, 8, 10, 12, 14, 16, 20, 24, 28, 30, 40, 42, 48, 56, 60, 70, 80, 84, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680

8 odd divisors

1, 3, 5, 7, 15, 21, 35, 105

How to compute the divisors of 1680?

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 1680 by each of the numbers from 1 to 1680 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)

1680 / 1 = 1680 (the remainder is 0, so 1 is a divisor of 1680)
1680 / 2 = 840 (the remainder is 0, so 2 is a divisor of 1680)
1680 / 3 = 560 (the remainder is 0, so 3 is a divisor of 1680)
...
1680 / 1679 = 1.0005955926147 (the remainder is 1, so 1679 is not a divisor of 1680)
1680 / 1680 = 1 (the remainder is 0, so 1680 is a divisor of 1680)

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 1680 (i.e. 40.987803063838). 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 $\frac{N}{d} = D$ )

1680 / 1 = 1680 (the remainder is 0, so 1 and 1680 are divisors of 1680)
1680 / 2 = 840 (the remainder is 0, so 2 and 840 are divisors of 1680)
1680 / 3 = 560 (the remainder is 0, so 3 and 560 are divisors of 1680)
...
1680 / 39 = 43.076923076923 (the remainder is 3, so 39 is not a divisor of 1680)
1680 / 40 = 42 (the remainder is 0, so 40 and 42 are divisors of 1680)