What are the divisors of 3961?

1, 17, 233, 3961

There is a total of 4 positive divisors.
The sum of these divisors is 4212.
The arithmetic mean is 1053.

4 odd divisors

1, 17, 233, 3961

How to compute the divisors of 3961?

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 3961 by each of the numbers from 1 to 3961 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)

3961 / 1 = 3961 (the remainder is 0, so 1 is a divisor of 3961)
3961 / 2 = 1980.5 (the remainder is 1, so 2 is not a divisor of 3961)
3961 / 3 = 1320.3333333333 (the remainder is 1, so 3 is not a divisor of 3961)
...
3961 / 3960 = 1.0002525252525 (the remainder is 1, so 3960 is not a divisor of 3961)
3961 / 3961 = 1 (the remainder is 0, so 3961 is a divisor of 3961)

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 3961 (i.e. 62.936475910238). 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 $\frac{N}{d} = D$ )

3961 / 1 = 3961 (the remainder is 0, so 1 and 3961 are divisors of 3961)
3961 / 2 = 1980.5 (the remainder is 1, so 2 is not a divisor of 3961)
3961 / 3 = 1320.3333333333 (the remainder is 1, so 3 is not a divisor of 3961)
...
3961 / 61 = 64.934426229508 (the remainder is 57, so 61 is not a divisor of 3961)
3961 / 62 = 63.887096774194 (the remainder is 55, so 62 is not a divisor of 3961)