What are the divisors of 1891?

1, 31, 61, 1891

There is a total of 4 positive divisors.
The sum of these divisors is 1984.
The arithmetic mean is 496.

4 odd divisors

1, 31, 61, 1891

How to compute the divisors of 1891?

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 1891 by each of the numbers from 1 to 1891 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)

1891 / 1 = 1891 (the remainder is 0, so 1 is a divisor of 1891)
1891 / 2 = 945.5 (the remainder is 1, so 2 is not a divisor of 1891)
1891 / 3 = 630.33333333333 (the remainder is 1, so 3 is not a divisor of 1891)
...
1891 / 1890 = 1.0005291005291 (the remainder is 1, so 1890 is not a divisor of 1891)
1891 / 1891 = 1 (the remainder is 0, so 1891 is a divisor of 1891)

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 1891 (i.e. 43.485629810318). 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 $\frac{N}{d} = D$ )

1891 / 1 = 1891 (the remainder is 0, so 1 and 1891 are divisors of 1891)
1891 / 2 = 945.5 (the remainder is 1, so 2 is not a divisor of 1891)
1891 / 3 = 630.33333333333 (the remainder is 1, so 3 is not a divisor of 1891)
...
1891 / 42 = 45.02380952381 (the remainder is 1, so 42 is not a divisor of 1891)
1891 / 43 = 43.976744186047 (the remainder is 42, so 43 is not a divisor of 1891)