What are the divisors of 1395?

1, 3, 5, 9, 15, 31, 45, 93, 155, 279, 465, 1395

There is a total of 12 positive divisors.
The sum of these divisors is 2496.
The arithmetic mean is 208.

12 odd divisors

1, 3, 5, 9, 15, 31, 45, 93, 155, 279, 465, 1395

How to compute the divisors of 1395?

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 1395 by each of the numbers from 1 to 1395 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)

1395 / 1 = 1395 (the remainder is 0, so 1 is a divisor of 1395)
1395 / 2 = 697.5 (the remainder is 1, so 2 is not a divisor of 1395)
1395 / 3 = 465 (the remainder is 0, so 3 is a divisor of 1395)
...
1395 / 1394 = 1.0007173601148 (the remainder is 1, so 1394 is not a divisor of 1395)
1395 / 1395 = 1 (the remainder is 0, so 1395 is a divisor of 1395)

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 1395 (i.e. 37.349698793966). 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 $\frac{N}{d} = D$ )

1395 / 1 = 1395 (the remainder is 0, so 1 and 1395 are divisors of 1395)
1395 / 2 = 697.5 (the remainder is 1, so 2 is not a divisor of 1395)
1395 / 3 = 465 (the remainder is 0, so 3 and 465 are divisors of 1395)
...
1395 / 36 = 38.75 (the remainder is 27, so 36 is not a divisor of 1395)
1395 / 37 = 37.702702702703 (the remainder is 26, so 37 is not a divisor of 1395)