What are the divisors of 1996?

1, 2, 4, 499, 998, 1996

There is a total of 6 positive divisors.
The sum of these divisors is 3500.
The arithmetic mean is 583.33333333333.

4 even divisors

2, 4, 998, 1996

2 odd divisors

1, 499

How to compute the divisors of 1996?

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 1996 by each of the numbers from 1 to 1996 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)

1996 / 1 = 1996 (the remainder is 0, so 1 is a divisor of 1996)
1996 / 2 = 998 (the remainder is 0, so 2 is a divisor of 1996)
1996 / 3 = 665.33333333333 (the remainder is 1, so 3 is not a divisor of 1996)
...
1996 / 1995 = 1.0005012531328 (the remainder is 1, so 1995 is not a divisor of 1996)
1996 / 1996 = 1 (the remainder is 0, so 1996 is a divisor of 1996)

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 1996 (i.e. 44.676615807377). 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 $\frac{N}{d} = D$ )

1996 / 1 = 1996 (the remainder is 0, so 1 and 1996 are divisors of 1996)
1996 / 2 = 998 (the remainder is 0, so 2 and 998 are divisors of 1996)
1996 / 3 = 665.33333333333 (the remainder is 1, so 3 is not a divisor of 1996)
...
1996 / 43 = 46.418604651163 (the remainder is 18, so 43 is not a divisor of 1996)
1996 / 44 = 45.363636363636 (the remainder is 16, so 44 is not a divisor of 1996)