What are the divisors of 1650?

1, 2, 3, 5, 6, 10, 11, 15, 22, 25, 30, 33, 50, 55, 66, 75, 110, 150, 165, 275, 330, 550, 825, 1650

There is a total of 24 positive divisors.
The sum of these divisors is 4464.
The arithmetic mean is 186.

12 even divisors

2, 6, 10, 22, 30, 50, 66, 110, 150, 330, 550, 1650

12 odd divisors

1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825

How to compute the divisors of 1650?

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 1650 by each of the numbers from 1 to 1650 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)

1650 / 1 = 1650 (the remainder is 0, so 1 is a divisor of 1650)
1650 / 2 = 825 (the remainder is 0, so 2 is a divisor of 1650)
1650 / 3 = 550 (the remainder is 0, so 3 is a divisor of 1650)
...
1650 / 1649 = 1.0006064281383 (the remainder is 1, so 1649 is not a divisor of 1650)
1650 / 1650 = 1 (the remainder is 0, so 1650 is a divisor of 1650)

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 1650 (i.e. 40.62019202318). 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 $\frac{N}{d} = D$ )

1650 / 1 = 1650 (the remainder is 0, so 1 and 1650 are divisors of 1650)
1650 / 2 = 825 (the remainder is 0, so 2 and 825 are divisors of 1650)
1650 / 3 = 550 (the remainder is 0, so 3 and 550 are divisors of 1650)
...
1650 / 39 = 42.307692307692 (the remainder is 12, so 39 is not a divisor of 1650)
1650 / 40 = 41.25 (the remainder is 10, so 40 is not a divisor of 1650)