What are the divisors of 3728?

1, 2, 4, 8, 16, 233, 466, 932, 1864, 3728

There is a total of 10 positive divisors.
The sum of these divisors is 7254.
The arithmetic mean is 725.4.

8 even divisors

2, 4, 8, 16, 466, 932, 1864, 3728

2 odd divisors

1, 233

How to compute the divisors of 3728?

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 3728 by each of the numbers from 1 to 3728 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)

3728 / 1 = 3728 (the remainder is 0, so 1 is a divisor of 3728)
3728 / 2 = 1864 (the remainder is 0, so 2 is a divisor of 3728)
3728 / 3 = 1242.6666666667 (the remainder is 2, so 3 is not a divisor of 3728)
...
3728 / 3727 = 1.0002683123155 (the remainder is 1, so 3727 is not a divisor of 3728)
3728 / 3728 = 1 (the remainder is 0, so 3728 is a divisor of 3728)

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 3728 (i.e. 61.057350089895). 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 $\frac{N}{d} = D$ )

3728 / 1 = 3728 (the remainder is 0, so 1 and 3728 are divisors of 3728)
3728 / 2 = 1864 (the remainder is 0, so 2 and 1864 are divisors of 3728)
3728 / 3 = 1242.6666666667 (the remainder is 2, so 3 is not a divisor of 3728)
...
3728 / 60 = 62.133333333333 (the remainder is 8, so 60 is not a divisor of 3728)
3728 / 61 = 61.114754098361 (the remainder is 7, so 61 is not a divisor of 3728)