What are the divisors of 1872?

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 36, 39, 48, 52, 72, 78, 104, 117, 144, 156, 208, 234, 312, 468, 624, 936, 1872

There is a total of 30 positive divisors.
The sum of these divisors is 5642.
The arithmetic mean is 188.06666666667.

24 even divisors

2, 4, 6, 8, 12, 16, 18, 24, 26, 36, 48, 52, 72, 78, 104, 144, 156, 208, 234, 312, 468, 624, 936, 1872

6 odd divisors

1, 3, 9, 13, 39, 117

How to compute the divisors of 1872?

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 1872 by each of the numbers from 1 to 1872 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)

1872 / 1 = 1872 (the remainder is 0, so 1 is a divisor of 1872)
1872 / 2 = 936 (the remainder is 0, so 2 is a divisor of 1872)
1872 / 3 = 624 (the remainder is 0, so 3 is a divisor of 1872)
...
1872 / 1871 = 1.0005344735436 (the remainder is 1, so 1871 is not a divisor of 1872)
1872 / 1872 = 1 (the remainder is 0, so 1872 is a divisor of 1872)

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 1872 (i.e. 43.266615305568). 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 $\frac{N}{d} = D$ )

1872 / 1 = 1872 (the remainder is 0, so 1 and 1872 are divisors of 1872)
1872 / 2 = 936 (the remainder is 0, so 2 and 936 are divisors of 1872)
1872 / 3 = 624 (the remainder is 0, so 3 and 624 are divisors of 1872)
...
1872 / 42 = 44.571428571429 (the remainder is 24, so 42 is not a divisor of 1872)
1872 / 43 = 43.53488372093 (the remainder is 23, so 43 is not a divisor of 1872)