What are the divisors of 3456?

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 192, 216, 288, 384, 432, 576, 864, 1152, 1728, 3456

There is a total of 32 positive divisors.
The sum of these divisors is 10200.
The arithmetic mean is 318.75.

28 even divisors

2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 192, 216, 288, 384, 432, 576, 864, 1152, 1728, 3456

4 odd divisors

1, 3, 9, 27

How to compute the divisors of 3456?

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 3456 by each of the numbers from 1 to 3456 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)

3456 / 1 = 3456 (the remainder is 0, so 1 is a divisor of 3456)
3456 / 2 = 1728 (the remainder is 0, so 2 is a divisor of 3456)
3456 / 3 = 1152 (the remainder is 0, so 3 is a divisor of 3456)
...
3456 / 3455 = 1.0002894356006 (the remainder is 1, so 3455 is not a divisor of 3456)
3456 / 3456 = 1 (the remainder is 0, so 3456 is a divisor of 3456)

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 3456 (i.e. 58.787753826796). 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 $\frac{N}{d} = D$ )

3456 / 1 = 3456 (the remainder is 0, so 1 and 3456 are divisors of 3456)
3456 / 2 = 1728 (the remainder is 0, so 2 and 1728 are divisors of 3456)
3456 / 3 = 1152 (the remainder is 0, so 3 and 1152 are divisors of 3456)
...
3456 / 57 = 60.631578947368 (the remainder is 36, so 57 is not a divisor of 3456)
3456 / 58 = 59.586206896552 (the remainder is 34, so 58 is not a divisor of 3456)