What are the divisors of 1131?

1, 3, 13, 29, 39, 87, 377, 1131

There is a total of 8 positive divisors.
The sum of these divisors is 1680.
The arithmetic mean is 210.

8 odd divisors

1, 3, 13, 29, 39, 87, 377, 1131

How to compute the divisors of 1131?

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 1131 by each of the numbers from 1 to 1131 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)

1131 / 1 = 1131 (the remainder is 0, so 1 is a divisor of 1131)
1131 / 2 = 565.5 (the remainder is 1, so 2 is not a divisor of 1131)
1131 / 3 = 377 (the remainder is 0, so 3 is a divisor of 1131)
...
1131 / 1130 = 1.0008849557522 (the remainder is 1, so 1130 is not a divisor of 1131)
1131 / 1131 = 1 (the remainder is 0, so 1131 is a divisor of 1131)

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 1131 (i.e. 33.6303434416). 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 $\frac{N}{d} = D$ )

1131 / 1 = 1131 (the remainder is 0, so 1 and 1131 are divisors of 1131)
1131 / 2 = 565.5 (the remainder is 1, so 2 is not a divisor of 1131)
1131 / 3 = 377 (the remainder is 0, so 3 and 377 are divisors of 1131)
...
1131 / 32 = 35.34375 (the remainder is 11, so 32 is not a divisor of 1131)
1131 / 33 = 34.272727272727 (the remainder is 9, so 33 is not a divisor of 1131)