What are the divisors of 1931?

1, 1931

There is a total of 2 positive divisors.
The sum of these divisors is 1932.
The arithmetic mean is 966.

2 odd divisors

1, 1931

How to compute the divisors of 1931?

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 1931 by each of the numbers from 1 to 1931 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)

1931 / 1 = 1931 (the remainder is 0, so 1 is a divisor of 1931)
1931 / 2 = 965.5 (the remainder is 1, so 2 is not a divisor of 1931)
1931 / 3 = 643.66666666667 (the remainder is 2, so 3 is not a divisor of 1931)
...
1931 / 1930 = 1.000518134715 (the remainder is 1, so 1930 is not a divisor of 1931)
1931 / 1931 = 1 (the remainder is 0, so 1931 is a divisor of 1931)

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 1931 (i.e. 43.94314508544). 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 $\frac{N}{d} = D$ )

1931 / 1 = 1931 (the remainder is 0, so 1 and 1931 are divisors of 1931)
1931 / 2 = 965.5 (the remainder is 1, so 2 is not a divisor of 1931)
1931 / 3 = 643.66666666667 (the remainder is 2, so 3 is not a divisor of 1931)
...
1931 / 42 = 45.97619047619 (the remainder is 41, so 42 is not a divisor of 1931)
1931 / 43 = 44.906976744186 (the remainder is 39, so 43 is not a divisor of 1931)