What are the divisors of 1986?

1, 2, 3, 6, 331, 662, 993, 1986

There is a total of 8 positive divisors.
The sum of these divisors is 3984.
The arithmetic mean is 498.

4 even divisors

2, 6, 662, 1986

4 odd divisors

1, 3, 331, 993

How to compute the divisors of 1986?

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 1986 by each of the numbers from 1 to 1986 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)

1986 / 1 = 1986 (the remainder is 0, so 1 is a divisor of 1986)
1986 / 2 = 993 (the remainder is 0, so 2 is a divisor of 1986)
1986 / 3 = 662 (the remainder is 0, so 3 is a divisor of 1986)
...
1986 / 1985 = 1.0005037783375 (the remainder is 1, so 1985 is not a divisor of 1986)
1986 / 1986 = 1 (the remainder is 0, so 1986 is a divisor of 1986)

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 1986 (i.e. 44.564559910314). 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 $\frac{N}{d} = D$ )

1986 / 1 = 1986 (the remainder is 0, so 1 and 1986 are divisors of 1986)
1986 / 2 = 993 (the remainder is 0, so 2 and 993 are divisors of 1986)
1986 / 3 = 662 (the remainder is 0, so 3 and 662 are divisors of 1986)
...
1986 / 43 = 46.186046511628 (the remainder is 8, so 43 is not a divisor of 1986)
1986 / 44 = 45.136363636364 (the remainder is 6, so 44 is not a divisor of 1986)