What are the divisors of 3696?

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 16, 21, 22, 24, 28, 33, 42, 44, 48, 56, 66, 77, 84, 88, 112, 132, 154, 168, 176, 231, 264, 308, 336, 462, 528, 616, 924, 1232, 1848, 3696

There is a total of 40 positive divisors.
The sum of these divisors is 11904.
The arithmetic mean is 297.6.

32 even divisors

2, 4, 6, 8, 12, 14, 16, 22, 24, 28, 42, 44, 48, 56, 66, 84, 88, 112, 132, 154, 168, 176, 264, 308, 336, 462, 528, 616, 924, 1232, 1848, 3696

8 odd divisors

1, 3, 7, 11, 21, 33, 77, 231

How to compute the divisors of 3696?

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 3696 by each of the numbers from 1 to 3696 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)

3696 / 1 = 3696 (the remainder is 0, so 1 is a divisor of 3696)
3696 / 2 = 1848 (the remainder is 0, so 2 is a divisor of 3696)
3696 / 3 = 1232 (the remainder is 0, so 3 is a divisor of 3696)
...
3696 / 3695 = 1.0002706359946 (the remainder is 1, so 3695 is not a divisor of 3696)
3696 / 3696 = 1 (the remainder is 0, so 3696 is a divisor of 3696)

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 3696 (i.e. 60.794736614283). 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 $\frac{N}{d} = D$ )

3696 / 1 = 3696 (the remainder is 0, so 1 and 3696 are divisors of 3696)
3696 / 2 = 1848 (the remainder is 0, so 2 and 1848 are divisors of 3696)
3696 / 3 = 1232 (the remainder is 0, so 3 and 1232 are divisors of 3696)
...
3696 / 59 = 62.64406779661 (the remainder is 38, so 59 is not a divisor of 3696)
3696 / 60 = 61.6 (the remainder is 36, so 60 is not a divisor of 3696)