What are the divisors of 3931?

1, 3931

There is a total of 2 positive divisors.
The sum of these divisors is 3932.
The arithmetic mean is 1966.

2 odd divisors

1, 3931

How to compute the divisors of 3931?

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 3931 by each of the numbers from 1 to 3931 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)

3931 / 1 = 3931 (the remainder is 0, so 1 is a divisor of 3931)
3931 / 2 = 1965.5 (the remainder is 1, so 2 is not a divisor of 3931)
3931 / 3 = 1310.3333333333 (the remainder is 1, so 3 is not a divisor of 3931)
...
3931 / 3930 = 1.0002544529262 (the remainder is 1, so 3930 is not a divisor of 3931)
3931 / 3931 = 1 (the remainder is 0, so 3931 is a divisor of 3931)

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 3931 (i.e. 62.697687357669). 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 $\frac{N}{d} = D$ )

3931 / 1 = 3931 (the remainder is 0, so 1 and 3931 are divisors of 3931)
3931 / 2 = 1965.5 (the remainder is 1, so 2 is not a divisor of 3931)
3931 / 3 = 1310.3333333333 (the remainder is 1, so 3 is not a divisor of 3931)
...
3931 / 61 = 64.44262295082 (the remainder is 27, so 61 is not a divisor of 3931)
3931 / 62 = 63.403225806452 (the remainder is 25, so 62 is not a divisor of 3931)