What are the divisors of 270?

1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270

There is a total of 16 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 45.

8 even divisors

2, 6, 10, 18, 30, 54, 90, 270

8 odd divisors

1, 3, 5, 9, 15, 27, 45, 135

How to compute the divisors of 270?

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 270 by each of the numbers from 1 to 270 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)

270 / 1 = 270 (the remainder is 0, so 1 is a divisor of 270)
270 / 2 = 135 (the remainder is 0, so 2 is a divisor of 270)
270 / 3 = 90 (the remainder is 0, so 3 is a divisor of 270)
...
270 / 269 = 1.003717472119 (the remainder is 1, so 269 is not a divisor of 270)
270 / 270 = 1 (the remainder is 0, so 270 is a divisor of 270)

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 270 (i.e. 16.431676725155). 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 $\frac{N}{d} = D$ )

270 / 1 = 270 (the remainder is 0, so 1 and 270 are divisors of 270)
270 / 2 = 135 (the remainder is 0, so 2 and 135 are divisors of 270)
270 / 3 = 90 (the remainder is 0, so 3 and 90 are divisors of 270)
...
270 / 15 = 18 (the remainder is 0, so 15 and 18 are divisors of 270)
270 / 16 = 16.875 (the remainder is 14, so 16 is not a divisor of 270)