What are the divisors of 3981?

1, 3, 1327, 3981

There is a total of 4 positive divisors.
The sum of these divisors is 5312.
The arithmetic mean is 1328.

4 odd divisors

1, 3, 1327, 3981

How to compute the divisors of 3981?

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 3981 by each of the numbers from 1 to 3981 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)

3981 / 1 = 3981 (the remainder is 0, so 1 is a divisor of 3981)
3981 / 2 = 1990.5 (the remainder is 1, so 2 is not a divisor of 3981)
3981 / 3 = 1327 (the remainder is 0, so 3 is a divisor of 3981)
...
3981 / 3980 = 1.0002512562814 (the remainder is 1, so 3980 is not a divisor of 3981)
3981 / 3981 = 1 (the remainder is 0, so 3981 is a divisor of 3981)

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 3981 (i.e. 63.095166217389). 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 $\frac{N}{d} = D$ )

3981 / 1 = 3981 (the remainder is 0, so 1 and 3981 are divisors of 3981)
3981 / 2 = 1990.5 (the remainder is 1, so 2 is not a divisor of 3981)
3981 / 3 = 1327 (the remainder is 0, so 3 and 1327 are divisors of 3981)
...
3981 / 62 = 64.209677419355 (the remainder is 13, so 62 is not a divisor of 3981)
3981 / 63 = 63.190476190476 (the remainder is 12, so 63 is not a divisor of 3981)