What are the divisors of 1896?

1, 2, 3, 4, 6, 8, 12, 24, 79, 158, 237, 316, 474, 632, 948, 1896

There is a total of 16 positive divisors.
The sum of these divisors is 4800.
The arithmetic mean is 300.

12 even divisors

2, 4, 6, 8, 12, 24, 158, 316, 474, 632, 948, 1896

4 odd divisors

1, 3, 79, 237

How to compute the divisors of 1896?

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 1896 by each of the numbers from 1 to 1896 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)

1896 / 1 = 1896 (the remainder is 0, so 1 is a divisor of 1896)
1896 / 2 = 948 (the remainder is 0, so 2 is a divisor of 1896)
1896 / 3 = 632 (the remainder is 0, so 3 is a divisor of 1896)
...
1896 / 1895 = 1.0005277044855 (the remainder is 1, so 1895 is not a divisor of 1896)
1896 / 1896 = 1 (the remainder is 0, so 1896 is a divisor of 1896)

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 1896 (i.e. 43.543082114154). 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 $\frac{N}{d} = D$ )

1896 / 1 = 1896 (the remainder is 0, so 1 and 1896 are divisors of 1896)
1896 / 2 = 948 (the remainder is 0, so 2 and 948 are divisors of 1896)
1896 / 3 = 632 (the remainder is 0, so 3 and 632 are divisors of 1896)
...
1896 / 42 = 45.142857142857 (the remainder is 6, so 42 is not a divisor of 1896)
1896 / 43 = 44.093023255814 (the remainder is 4, so 43 is not a divisor of 1896)