What are the divisors of 3966?

1, 2, 3, 6, 661, 1322, 1983, 3966

There is a total of 8 positive divisors.
The sum of these divisors is 7944.
The arithmetic mean is 993.

4 even divisors

2, 6, 1322, 3966

4 odd divisors

1, 3, 661, 1983

How to compute the divisors of 3966?

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 3966 by each of the numbers from 1 to 3966 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)

3966 / 1 = 3966 (the remainder is 0, so 1 is a divisor of 3966)
3966 / 2 = 1983 (the remainder is 0, so 2 is a divisor of 3966)
3966 / 3 = 1322 (the remainder is 0, so 3 is a divisor of 3966)
...
3966 / 3965 = 1.0002522068096 (the remainder is 1, so 3965 is not a divisor of 3966)
3966 / 3966 = 1 (the remainder is 0, so 3966 is a divisor of 3966)

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 3966 (i.e. 62.976185975335). 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 $\frac{N}{d} = D$ )

3966 / 1 = 3966 (the remainder is 0, so 1 and 3966 are divisors of 3966)
3966 / 2 = 1983 (the remainder is 0, so 2 and 1983 are divisors of 3966)
3966 / 3 = 1322 (the remainder is 0, so 3 and 1322 are divisors of 3966)
...
3966 / 61 = 65.016393442623 (the remainder is 1, so 61 is not a divisor of 3966)
3966 / 62 = 63.967741935484 (the remainder is 60, so 62 is not a divisor of 3966)