What are the divisors of 1365?

1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365

There is a total of 16 positive divisors.
The sum of these divisors is 2688.
The arithmetic mean is 168.

16 odd divisors

1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365

How to compute the divisors of 1365?

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 1365 by each of the numbers from 1 to 1365 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)

1365 / 1 = 1365 (the remainder is 0, so 1 is a divisor of 1365)
1365 / 2 = 682.5 (the remainder is 1, so 2 is not a divisor of 1365)
1365 / 3 = 455 (the remainder is 0, so 3 is a divisor of 1365)
...
1365 / 1364 = 1.0007331378299 (the remainder is 1, so 1364 is not a divisor of 1365)
1365 / 1365 = 1 (the remainder is 0, so 1365 is a divisor of 1365)

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 1365 (i.e. 36.945906403822). 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 $\frac{N}{d} = D$ )

1365 / 1 = 1365 (the remainder is 0, so 1 and 1365 are divisors of 1365)
1365 / 2 = 682.5 (the remainder is 1, so 2 is not a divisor of 1365)
1365 / 3 = 455 (the remainder is 0, so 3 and 455 are divisors of 1365)
...
1365 / 35 = 39 (the remainder is 0, so 35 and 39 are divisors of 1365)
1365 / 36 = 37.916666666667 (the remainder is 33, so 36 is not a divisor of 1365)