What are the divisors of 1215?

1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 1215

There is a total of 12 positive divisors.
The sum of these divisors is 2184.
The arithmetic mean is 182.

12 odd divisors

1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 1215

How to compute the divisors of 1215?

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 1215 by each of the numbers from 1 to 1215 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)

1215 / 1 = 1215 (the remainder is 0, so 1 is a divisor of 1215)
1215 / 2 = 607.5 (the remainder is 1, so 2 is not a divisor of 1215)
1215 / 3 = 405 (the remainder is 0, so 3 is a divisor of 1215)
...
1215 / 1214 = 1.000823723229 (the remainder is 1, so 1214 is not a divisor of 1215)
1215 / 1215 = 1 (the remainder is 0, so 1215 is a divisor of 1215)

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 1215 (i.e. 34.856850115867). 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 $\frac{N}{d} = D$ )

1215 / 1 = 1215 (the remainder is 0, so 1 and 1215 are divisors of 1215)
1215 / 2 = 607.5 (the remainder is 1, so 2 is not a divisor of 1215)
1215 / 3 = 405 (the remainder is 0, so 3 and 405 are divisors of 1215)
...
1215 / 33 = 36.818181818182 (the remainder is 27, so 33 is not a divisor of 1215)
1215 / 34 = 35.735294117647 (the remainder is 25, so 34 is not a divisor of 1215)