What are the divisors of 3984?

1, 2, 3, 4, 6, 8, 12, 16, 24, 48, 83, 166, 249, 332, 498, 664, 996, 1328, 1992, 3984

There is a total of 20 positive divisors.
The sum of these divisors is 10416.
The arithmetic mean is 520.8.

16 even divisors

2, 4, 6, 8, 12, 16, 24, 48, 166, 332, 498, 664, 996, 1328, 1992, 3984

4 odd divisors

1, 3, 83, 249

How to compute the divisors of 3984?

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 3984 by each of the numbers from 1 to 3984 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)

3984 / 1 = 3984 (the remainder is 0, so 1 is a divisor of 3984)
3984 / 2 = 1992 (the remainder is 0, so 2 is a divisor of 3984)
3984 / 3 = 1328 (the remainder is 0, so 3 is a divisor of 3984)
...
3984 / 3983 = 1.0002510670349 (the remainder is 1, so 3983 is not a divisor of 3984)
3984 / 3984 = 1 (the remainder is 0, so 3984 is a divisor of 3984)

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 3984 (i.e. 63.118935352238). 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 $\frac{N}{d} = D$ )

3984 / 1 = 3984 (the remainder is 0, so 1 and 3984 are divisors of 3984)
3984 / 2 = 1992 (the remainder is 0, so 2 and 1992 are divisors of 3984)
3984 / 3 = 1328 (the remainder is 0, so 3 and 1328 are divisors of 3984)
...
3984 / 62 = 64.258064516129 (the remainder is 16, so 62 is not a divisor of 3984)
3984 / 63 = 63.238095238095 (the remainder is 15, so 63 is not a divisor of 3984)