What are the divisors of 4096?

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096

There is a total of 13 positive divisors.
The sum of these divisors is 8191.
The arithmetic mean is 630.07692307692.

12 even divisors

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096

1 odd divisors

How to compute the divisors of 4096?

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 4096 by each of the numbers from 1 to 4096 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)

4096 / 1 = 4096 (the remainder is 0, so 1 is a divisor of 4096)
4096 / 2 = 2048 (the remainder is 0, so 2 is a divisor of 4096)
4096 / 3 = 1365.3333333333 (the remainder is 1, so 3 is not a divisor of 4096)
...
4096 / 4095 = 1.0002442002442 (the remainder is 1, so 4095 is not a divisor of 4096)
4096 / 4096 = 1 (the remainder is 0, so 4096 is a divisor of 4096)

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 4096 (i.e. 64). 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 $\frac{N}{d} = D$ )

4096 / 1 = 4096 (the remainder is 0, so 1 and 4096 are divisors of 4096)
4096 / 2 = 2048 (the remainder is 0, so 2 and 2048 are divisors of 4096)
4096 / 3 = 1365.3333333333 (the remainder is 1, so 3 is not a divisor of 4096)
...
4096 / 63 = 65.015873015873 (the remainder is 1, so 63 is not a divisor of 4096)
4096 / 64 = 64 (the remainder is 0, so 64 and 64 are divisors of 4096)