What are the divisors of 1911?

1, 3, 7, 13, 21, 39, 49, 91, 147, 273, 637, 1911

There is a total of 12 positive divisors.
The sum of these divisors is 3192.
The arithmetic mean is 266.

12 odd divisors

1, 3, 7, 13, 21, 39, 49, 91, 147, 273, 637, 1911

How to compute the divisors of 1911?

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 1911 by each of the numbers from 1 to 1911 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)

1911 / 1 = 1911 (the remainder is 0, so 1 is a divisor of 1911)
1911 / 2 = 955.5 (the remainder is 1, so 2 is not a divisor of 1911)
1911 / 3 = 637 (the remainder is 0, so 3 is a divisor of 1911)
...
1911 / 1910 = 1.0005235602094 (the remainder is 1, so 1910 is not a divisor of 1911)
1911 / 1911 = 1 (the remainder is 0, so 1911 is a divisor of 1911)

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 1911 (i.e. 43.714985988789). 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 $\frac{N}{d} = D$ )

1911 / 1 = 1911 (the remainder is 0, so 1 and 1911 are divisors of 1911)
1911 / 2 = 955.5 (the remainder is 1, so 2 is not a divisor of 1911)
1911 / 3 = 637 (the remainder is 0, so 3 and 637 are divisors of 1911)
...
1911 / 42 = 45.5 (the remainder is 21, so 42 is not a divisor of 1911)
1911 / 43 = 44.441860465116 (the remainder is 19, so 43 is not a divisor of 1911)