What are the divisors of 3993?

1, 3, 11, 33, 121, 363, 1331, 3993

There is a total of 8 positive divisors.
The sum of these divisors is 5856.
The arithmetic mean is 732.

8 odd divisors

1, 3, 11, 33, 121, 363, 1331, 3993

How to compute the divisors of 3993?

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 3993 by each of the numbers from 1 to 3993 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)

3993 / 1 = 3993 (the remainder is 0, so 1 is a divisor of 3993)
3993 / 2 = 1996.5 (the remainder is 1, so 2 is not a divisor of 3993)
3993 / 3 = 1331 (the remainder is 0, so 3 is a divisor of 3993)
...
3993 / 3992 = 1.000250501002 (the remainder is 1, so 3992 is not a divisor of 3993)
3993 / 3993 = 1 (the remainder is 0, so 3993 is a divisor of 3993)

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 3993 (i.e. 63.190189111918). 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 $\frac{N}{d} = D$ )

3993 / 1 = 3993 (the remainder is 0, so 1 and 3993 are divisors of 3993)
3993 / 2 = 1996.5 (the remainder is 1, so 2 is not a divisor of 3993)
3993 / 3 = 1331 (the remainder is 0, so 3 and 1331 are divisors of 3993)
...
3993 / 62 = 64.403225806452 (the remainder is 25, so 62 is not a divisor of 3993)
3993 / 63 = 63.380952380952 (the remainder is 24, so 63 is not a divisor of 3993)