What are the divisors of 1985?

1, 5, 397, 1985

There is a total of 4 positive divisors.
The sum of these divisors is 2388.
The arithmetic mean is 597.

4 odd divisors

1, 5, 397, 1985

How to compute the divisors of 1985?

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 1985 by each of the numbers from 1 to 1985 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)

1985 / 1 = 1985 (the remainder is 0, so 1 is a divisor of 1985)
1985 / 2 = 992.5 (the remainder is 1, so 2 is not a divisor of 1985)
1985 / 3 = 661.66666666667 (the remainder is 2, so 3 is not a divisor of 1985)
...
1985 / 1984 = 1.0005040322581 (the remainder is 1, so 1984 is not a divisor of 1985)
1985 / 1985 = 1 (the remainder is 0, so 1985 is a divisor of 1985)

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 1985 (i.e. 44.553338819891). 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 $\frac{N}{d} = D$ )

1985 / 1 = 1985 (the remainder is 0, so 1 and 1985 are divisors of 1985)
1985 / 2 = 992.5 (the remainder is 1, so 2 is not a divisor of 1985)
1985 / 3 = 661.66666666667 (the remainder is 2, so 3 is not a divisor of 1985)
...
1985 / 43 = 46.162790697674 (the remainder is 7, so 43 is not a divisor of 1985)
1985 / 44 = 45.113636363636 (the remainder is 5, so 44 is not a divisor of 1985)