What are the divisors of 1994?

1, 2, 997, 1994

There is a total of 4 positive divisors.
The sum of these divisors is 2994.
The arithmetic mean is 748.5.

2 even divisors

2, 1994

2 odd divisors

1, 997

How to compute the divisors of 1994?

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 1994 by each of the numbers from 1 to 1994 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)

1994 / 1 = 1994 (the remainder is 0, so 1 is a divisor of 1994)
1994 / 2 = 997 (the remainder is 0, so 2 is a divisor of 1994)
1994 / 3 = 664.66666666667 (the remainder is 2, so 3 is not a divisor of 1994)
...
1994 / 1993 = 1.0005017561465 (the remainder is 1, so 1993 is not a divisor of 1994)
1994 / 1994 = 1 (the remainder is 0, so 1994 is a divisor of 1994)

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 1994 (i.e. 44.654227123532). 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 $\frac{N}{d} = D$ )

1994 / 1 = 1994 (the remainder is 0, so 1 and 1994 are divisors of 1994)
1994 / 2 = 997 (the remainder is 0, so 2 and 997 are divisors of 1994)
1994 / 3 = 664.66666666667 (the remainder is 2, so 3 is not a divisor of 1994)
...
1994 / 43 = 46.372093023256 (the remainder is 16, so 43 is not a divisor of 1994)
1994 / 44 = 45.318181818182 (the remainder is 14, so 44 is not a divisor of 1994)