What are the divisors of 1350?

1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350

There is a total of 24 positive divisors.
The sum of these divisors is 3720.
The arithmetic mean is 155.

12 even divisors

2, 6, 10, 18, 30, 50, 54, 90, 150, 270, 450, 1350

12 odd divisors

1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675

How to compute the divisors of 1350?

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 1350 by each of the numbers from 1 to 1350 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)

1350 / 1 = 1350 (the remainder is 0, so 1 is a divisor of 1350)
1350 / 2 = 675 (the remainder is 0, so 2 is a divisor of 1350)
1350 / 3 = 450 (the remainder is 0, so 3 is a divisor of 1350)
...
1350 / 1349 = 1.0007412898443 (the remainder is 1, so 1349 is not a divisor of 1350)
1350 / 1350 = 1 (the remainder is 0, so 1350 is a divisor of 1350)

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 1350 (i.e. 36.742346141748). 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 $\frac{N}{d} = D$ )

1350 / 1 = 1350 (the remainder is 0, so 1 and 1350 are divisors of 1350)
1350 / 2 = 675 (the remainder is 0, so 2 and 675 are divisors of 1350)
1350 / 3 = 450 (the remainder is 0, so 3 and 450 are divisors of 1350)
...
1350 / 35 = 38.571428571429 (the remainder is 20, so 35 is not a divisor of 1350)
1350 / 36 = 37.5 (the remainder is 18, so 36 is not a divisor of 1350)