What are the divisors of 1335?

1, 3, 5, 15, 89, 267, 445, 1335

There is a total of 8 positive divisors.
The sum of these divisors is 2160.
The arithmetic mean is 270.

8 odd divisors

1, 3, 5, 15, 89, 267, 445, 1335

How to compute the divisors of 1335?

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 1335 by each of the numbers from 1 to 1335 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)

1335 / 1 = 1335 (the remainder is 0, so 1 is a divisor of 1335)
1335 / 2 = 667.5 (the remainder is 1, so 2 is not a divisor of 1335)
1335 / 3 = 445 (the remainder is 0, so 3 is a divisor of 1335)
...
1335 / 1334 = 1.0007496251874 (the remainder is 1, so 1334 is not a divisor of 1335)
1335 / 1335 = 1 (the remainder is 0, so 1335 is a divisor of 1335)

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 1335 (i.e. 36.53765181289). 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 $\frac{N}{d} = D$ )

1335 / 1 = 1335 (the remainder is 0, so 1 and 1335 are divisors of 1335)
1335 / 2 = 667.5 (the remainder is 1, so 2 is not a divisor of 1335)
1335 / 3 = 445 (the remainder is 0, so 3 and 445 are divisors of 1335)
...
1335 / 35 = 38.142857142857 (the remainder is 5, so 35 is not a divisor of 1335)
1335 / 36 = 37.083333333333 (the remainder is 3, so 36 is not a divisor of 1335)