What are the divisors of 993?

1, 3, 331, 993

There is a total of 4 positive divisors.
The sum of these divisors is 1328.
The arithmetic mean is 332.

4 odd divisors

1, 3, 331, 993

How to compute the divisors of 993?

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 993 by each of the numbers from 1 to 993 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)

993 / 1 = 993 (the remainder is 0, so 1 is a divisor of 993)
993 / 2 = 496.5 (the remainder is 1, so 2 is not a divisor of 993)
993 / 3 = 331 (the remainder is 0, so 3 is a divisor of 993)
...
993 / 992 = 1.0010080645161 (the remainder is 1, so 992 is not a divisor of 993)
993 / 993 = 1 (the remainder is 0, so 993 is a divisor of 993)

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 993 (i.e. 31.511902513177). 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 $\frac{N}{d} = D$ )

993 / 1 = 993 (the remainder is 0, so 1 and 993 are divisors of 993)
993 / 2 = 496.5 (the remainder is 1, so 2 is not a divisor of 993)
993 / 3 = 331 (the remainder is 0, so 3 and 331 are divisors of 993)
...
993 / 30 = 33.1 (the remainder is 3, so 30 is not a divisor of 993)
993 / 31 = 32.032258064516 (the remainder is 1, so 31 is not a divisor of 993)