What are the divisors of 1993?

1, 1993

There is a total of 2 positive divisors.
The sum of these divisors is 1994.
The arithmetic mean is 997.

2 odd divisors

1, 1993

How to compute the divisors of 1993?

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 1993 by each of the numbers from 1 to 1993 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)

1993 / 1 = 1993 (the remainder is 0, so 1 is a divisor of 1993)
1993 / 2 = 996.5 (the remainder is 1, so 2 is not a divisor of 1993)
1993 / 3 = 664.33333333333 (the remainder is 1, so 3 is not a divisor of 1993)
...
1993 / 1992 = 1.0005020080321 (the remainder is 1, so 1992 is not a divisor of 1993)
1993 / 1993 = 1 (the remainder is 0, so 1993 is a divisor of 1993)

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 1993 (i.e. 44.643028571099). 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 $\frac{N}{d} = D$ )

1993 / 1 = 1993 (the remainder is 0, so 1 and 1993 are divisors of 1993)
1993 / 2 = 996.5 (the remainder is 1, so 2 is not a divisor of 1993)
1993 / 3 = 664.33333333333 (the remainder is 1, so 3 is not a divisor of 1993)
...
1993 / 43 = 46.348837209302 (the remainder is 15, so 43 is not a divisor of 1993)
1993 / 44 = 45.295454545455 (the remainder is 13, so 44 is not a divisor of 1993)