What are the divisors of 3968?

1, 2, 4, 8, 16, 31, 32, 62, 64, 124, 128, 248, 496, 992, 1984, 3968

There is a total of 16 positive divisors.
The sum of these divisors is 8160.
The arithmetic mean is 510.

14 even divisors

2, 4, 8, 16, 32, 62, 64, 124, 128, 248, 496, 992, 1984, 3968

2 odd divisors

1, 31

How to compute the divisors of 3968?

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 3968 by each of the numbers from 1 to 3968 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)

3968 / 1 = 3968 (the remainder is 0, so 1 is a divisor of 3968)
3968 / 2 = 1984 (the remainder is 0, so 2 is a divisor of 3968)
3968 / 3 = 1322.6666666667 (the remainder is 2, so 3 is not a divisor of 3968)
...
3968 / 3967 = 1.0002520796572 (the remainder is 1, so 3967 is not a divisor of 3968)
3968 / 3968 = 1 (the remainder is 0, so 3968 is a divisor of 3968)

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 3968 (i.e. 62.992062992094). 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 $\frac{N}{d} = D$ )

3968 / 1 = 3968 (the remainder is 0, so 1 and 3968 are divisors of 3968)
3968 / 2 = 1984 (the remainder is 0, so 2 and 1984 are divisors of 3968)
3968 / 3 = 1322.6666666667 (the remainder is 2, so 3 is not a divisor of 3968)
...
3968 / 61 = 65.049180327869 (the remainder is 3, so 61 is not a divisor of 3968)
3968 / 62 = 64 (the remainder is 0, so 62 and 64 are divisors of 3968)