What are the divisors of 450?

1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450

There is a total of 18 positive divisors.
The sum of these divisors is 1209.
The arithmetic mean is 67.166666666667.

9 even divisors

2, 6, 10, 18, 30, 50, 90, 150, 450

9 odd divisors

1, 3, 5, 9, 15, 25, 45, 75, 225

How to compute the divisors of 450?

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 450 by each of the numbers from 1 to 450 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)

450 / 1 = 450 (the remainder is 0, so 1 is a divisor of 450)
450 / 2 = 225 (the remainder is 0, so 2 is a divisor of 450)
450 / 3 = 150 (the remainder is 0, so 3 is a divisor of 450)
...
450 / 449 = 1.0022271714922 (the remainder is 1, so 449 is not a divisor of 450)
450 / 450 = 1 (the remainder is 0, so 450 is a divisor of 450)

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 450 (i.e. 21.213203435596). 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 $\frac{N}{d} = D$ )

450 / 1 = 450 (the remainder is 0, so 1 and 450 are divisors of 450)
450 / 2 = 225 (the remainder is 0, so 2 and 225 are divisors of 450)
450 / 3 = 150 (the remainder is 0, so 3 and 150 are divisors of 450)
...
450 / 20 = 22.5 (the remainder is 10, so 20 is not a divisor of 450)
450 / 21 = 21.428571428571 (the remainder is 9, so 21 is not a divisor of 450)