What are the divisors of 3998?

1, 2, 1999, 3998

There is a total of 4 positive divisors.
The sum of these divisors is 6000.
The arithmetic mean is 1500.

2 even divisors

2, 3998

2 odd divisors

1, 1999

How to compute the divisors of 3998?

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 3998 by each of the numbers from 1 to 3998 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)

3998 / 1 = 3998 (the remainder is 0, so 1 is a divisor of 3998)
3998 / 2 = 1999 (the remainder is 0, so 2 is a divisor of 3998)
3998 / 3 = 1332.6666666667 (the remainder is 2, so 3 is not a divisor of 3998)
...
3998 / 3997 = 1.0002501876407 (the remainder is 1, so 3997 is not a divisor of 3998)
3998 / 3998 = 1 (the remainder is 0, so 3998 is a divisor of 3998)

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 3998 (i.e. 63.229739838149). 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 $\frac{N}{d} = D$ )

3998 / 1 = 3998 (the remainder is 0, so 1 and 3998 are divisors of 3998)
3998 / 2 = 1999 (the remainder is 0, so 2 and 1999 are divisors of 3998)
3998 / 3 = 1332.6666666667 (the remainder is 2, so 3 is not a divisor of 3998)
...
3998 / 62 = 64.483870967742 (the remainder is 30, so 62 is not a divisor of 3998)
3998 / 63 = 63.460317460317 (the remainder is 29, so 63 is not a divisor of 3998)