What are the divisors of 3648?

1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 32, 38, 48, 57, 64, 76, 96, 114, 152, 192, 228, 304, 456, 608, 912, 1216, 1824, 3648

There is a total of 28 positive divisors.
The sum of these divisors is 10160.
The arithmetic mean is 362.85714285714.

24 even divisors

2, 4, 6, 8, 12, 16, 24, 32, 38, 48, 64, 76, 96, 114, 152, 192, 228, 304, 456, 608, 912, 1216, 1824, 3648

4 odd divisors

1, 3, 19, 57

How to compute the divisors of 3648?

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 3648 by each of the numbers from 1 to 3648 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)

3648 / 1 = 3648 (the remainder is 0, so 1 is a divisor of 3648)
3648 / 2 = 1824 (the remainder is 0, so 2 is a divisor of 3648)
3648 / 3 = 1216 (the remainder is 0, so 3 is a divisor of 3648)
...
3648 / 3647 = 1.0002741979709 (the remainder is 1, so 3647 is not a divisor of 3648)
3648 / 3648 = 1 (the remainder is 0, so 3648 is a divisor of 3648)

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 3648 (i.e. 60.398675482166). 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 $\frac{N}{d} = D$ )

3648 / 1 = 3648 (the remainder is 0, so 1 and 3648 are divisors of 3648)
3648 / 2 = 1824 (the remainder is 0, so 2 and 1824 are divisors of 3648)
3648 / 3 = 1216 (the remainder is 0, so 3 and 1216 are divisors of 3648)
...
3648 / 59 = 61.830508474576 (the remainder is 49, so 59 is not a divisor of 3648)
3648 / 60 = 60.8 (the remainder is 48, so 60 is not a divisor of 3648)