What are the divisors of 3937?

1, 31, 127, 3937

There is a total of 4 positive divisors.
The sum of these divisors is 4096.
The arithmetic mean is 1024.

4 odd divisors

1, 31, 127, 3937

How to compute the divisors of 3937?

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 3937 by each of the numbers from 1 to 3937 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)

3937 / 1 = 3937 (the remainder is 0, so 1 is a divisor of 3937)
3937 / 2 = 1968.5 (the remainder is 1, so 2 is not a divisor of 3937)
3937 / 3 = 1312.3333333333 (the remainder is 1, so 3 is not a divisor of 3937)
...
3937 / 3936 = 1.0002540650407 (the remainder is 1, so 3936 is not a divisor of 3937)
3937 / 3937 = 1 (the remainder is 0, so 3937 is a divisor of 3937)

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 3937 (i.e. 62.745517768204). 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 $\frac{N}{d} = D$ )

3937 / 1 = 3937 (the remainder is 0, so 1 and 3937 are divisors of 3937)
3937 / 2 = 1968.5 (the remainder is 1, so 2 is not a divisor of 3937)
3937 / 3 = 1312.3333333333 (the remainder is 1, so 3 is not a divisor of 3937)
...
3937 / 61 = 64.540983606557 (the remainder is 33, so 61 is not a divisor of 3937)
3937 / 62 = 63.5 (the remainder is 31, so 62 is not a divisor of 3937)