What are the divisors of 3971?

1, 11, 19, 209, 361, 3971

There is a total of 6 positive divisors.
The sum of these divisors is 4572.
The arithmetic mean is 762.

6 odd divisors

1, 11, 19, 209, 361, 3971

How to compute the divisors of 3971?

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 3971 by each of the numbers from 1 to 3971 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)

3971 / 1 = 3971 (the remainder is 0, so 1 is a divisor of 3971)
3971 / 2 = 1985.5 (the remainder is 1, so 2 is not a divisor of 3971)
3971 / 3 = 1323.6666666667 (the remainder is 2, so 3 is not a divisor of 3971)
...
3971 / 3970 = 1.0002518891688 (the remainder is 1, so 3970 is not a divisor of 3971)
3971 / 3971 = 1 (the remainder is 0, so 3971 is a divisor of 3971)

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 3971 (i.e. 63.015871016753). 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 $\frac{N}{d} = D$ )

3971 / 1 = 3971 (the remainder is 0, so 1 and 3971 are divisors of 3971)
3971 / 2 = 1985.5 (the remainder is 1, so 2 is not a divisor of 3971)
3971 / 3 = 1323.6666666667 (the remainder is 2, so 3 is not a divisor of 3971)
...
3971 / 62 = 64.048387096774 (the remainder is 3, so 62 is not a divisor of 3971)
3971 / 63 = 63.031746031746 (the remainder is 2, so 63 is not a divisor of 3971)