What are the divisors of 1950?

1, 2, 3, 5, 6, 10, 13, 15, 25, 26, 30, 39, 50, 65, 75, 78, 130, 150, 195, 325, 390, 650, 975, 1950

There is a total of 24 positive divisors.
The sum of these divisors is 5208.
The arithmetic mean is 217.

12 even divisors

2, 6, 10, 26, 30, 50, 78, 130, 150, 390, 650, 1950

12 odd divisors

1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975

How to compute the divisors of 1950?

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 1950 by each of the numbers from 1 to 1950 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)

1950 / 1 = 1950 (the remainder is 0, so 1 is a divisor of 1950)
1950 / 2 = 975 (the remainder is 0, so 2 is a divisor of 1950)
1950 / 3 = 650 (the remainder is 0, so 3 is a divisor of 1950)
...
1950 / 1949 = 1.0005130836326 (the remainder is 1, so 1949 is not a divisor of 1950)
1950 / 1950 = 1 (the remainder is 0, so 1950 is a divisor of 1950)

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 1950 (i.e. 44.158804331639). 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 $\frac{N}{d} = D$ )

1950 / 1 = 1950 (the remainder is 0, so 1 and 1950 are divisors of 1950)
1950 / 2 = 975 (the remainder is 0, so 2 and 975 are divisors of 1950)
1950 / 3 = 650 (the remainder is 0, so 3 and 650 are divisors of 1950)
...
1950 / 43 = 45.348837209302 (the remainder is 15, so 43 is not a divisor of 1950)
1950 / 44 = 44.318181818182 (the remainder is 14, so 44 is not a divisor of 1950)