What are the divisors of 1890?

1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 27, 30, 35, 42, 45, 54, 63, 70, 90, 105, 126, 135, 189, 210, 270, 315, 378, 630, 945, 1890

There is a total of 32 positive divisors.
The sum of these divisors is 5760.
The arithmetic mean is 180.

16 even divisors

2, 6, 10, 14, 18, 30, 42, 54, 70, 90, 126, 210, 270, 378, 630, 1890

16 odd divisors

1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945

How to compute the divisors of 1890?

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 1890 by each of the numbers from 1 to 1890 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)

1890 / 1 = 1890 (the remainder is 0, so 1 is a divisor of 1890)
1890 / 2 = 945 (the remainder is 0, so 2 is a divisor of 1890)
1890 / 3 = 630 (the remainder is 0, so 3 is a divisor of 1890)
...
1890 / 1889 = 1.0005293806247 (the remainder is 1, so 1889 is not a divisor of 1890)
1890 / 1890 = 1 (the remainder is 0, so 1890 is a divisor of 1890)

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 1890 (i.e. 43.474130238568). 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 $\frac{N}{d} = D$ )

1890 / 1 = 1890 (the remainder is 0, so 1 and 1890 are divisors of 1890)
1890 / 2 = 945 (the remainder is 0, so 2 and 945 are divisors of 1890)
1890 / 3 = 630 (the remainder is 0, so 3 and 630 are divisors of 1890)
...
1890 / 42 = 45 (the remainder is 0, so 42 and 45 are divisors of 1890)
1890 / 43 = 43.953488372093 (the remainder is 41, so 43 is not a divisor of 1890)