What are the divisors of 1331?

1, 11, 121, 1331

There is a total of 4 positive divisors.
The sum of these divisors is 1464.
The arithmetic mean is 366.

4 odd divisors

1, 11, 121, 1331

How to compute the divisors of 1331?

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 1331 by each of the numbers from 1 to 1331 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)

1331 / 1 = 1331 (the remainder is 0, so 1 is a divisor of 1331)
1331 / 2 = 665.5 (the remainder is 1, so 2 is not a divisor of 1331)
1331 / 3 = 443.66666666667 (the remainder is 2, so 3 is not a divisor of 1331)
...
1331 / 1330 = 1.0007518796992 (the remainder is 1, so 1330 is not a divisor of 1331)
1331 / 1331 = 1 (the remainder is 0, so 1331 is a divisor of 1331)

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 1331 (i.e. 36.482872693909). 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 $\frac{N}{d} = D$ )

1331 / 1 = 1331 (the remainder is 0, so 1 and 1331 are divisors of 1331)
1331 / 2 = 665.5 (the remainder is 1, so 2 is not a divisor of 1331)
1331 / 3 = 443.66666666667 (the remainder is 2, so 3 is not a divisor of 1331)
...
1331 / 35 = 38.028571428571 (the remainder is 1, so 35 is not a divisor of 1331)
1331 / 36 = 36.972222222222 (the remainder is 35, so 36 is not a divisor of 1331)