What are the divisors of 1311?

1, 3, 19, 23, 57, 69, 437, 1311

There is a total of 8 positive divisors.
The sum of these divisors is 1920.
The arithmetic mean is 240.

8 odd divisors

1, 3, 19, 23, 57, 69, 437, 1311

How to compute the divisors of 1311?

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 1311 by each of the numbers from 1 to 1311 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)

1311 / 1 = 1311 (the remainder is 0, so 1 is a divisor of 1311)
1311 / 2 = 655.5 (the remainder is 1, so 2 is not a divisor of 1311)
1311 / 3 = 437 (the remainder is 0, so 3 is a divisor of 1311)
...
1311 / 1310 = 1.0007633587786 (the remainder is 1, so 1310 is not a divisor of 1311)
1311 / 1311 = 1 (the remainder is 0, so 1311 is a divisor of 1311)

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 1311 (i.e. 36.207733980463). 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 $\frac{N}{d} = D$ )

1311 / 1 = 1311 (the remainder is 0, so 1 and 1311 are divisors of 1311)
1311 / 2 = 655.5 (the remainder is 1, so 2 is not a divisor of 1311)
1311 / 3 = 437 (the remainder is 0, so 3 and 437 are divisors of 1311)
...
1311 / 35 = 37.457142857143 (the remainder is 16, so 35 is not a divisor of 1311)
1311 / 36 = 36.416666666667 (the remainder is 15, so 36 is not a divisor of 1311)