What are the divisors of 2093?

1, 7, 13, 23, 91, 161, 299, 2093

There is a total of 8 positive divisors.
The sum of these divisors is 2688.
The arithmetic mean is 336.

8 odd divisors

1, 7, 13, 23, 91, 161, 299, 2093

How to compute the divisors of 2093?

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 2093 by each of the numbers from 1 to 2093 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)

2093 / 1 = 2093 (the remainder is 0, so 1 is a divisor of 2093)
2093 / 2 = 1046.5 (the remainder is 1, so 2 is not a divisor of 2093)
2093 / 3 = 697.66666666667 (the remainder is 2, so 3 is not a divisor of 2093)
...
2093 / 2092 = 1.0004780114723 (the remainder is 1, so 2092 is not a divisor of 2093)
2093 / 2093 = 1 (the remainder is 0, so 2093 is a divisor of 2093)

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 2093 (i.e. 45.749316934791). 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 $\frac{N}{d} = D$ )

2093 / 1 = 2093 (the remainder is 0, so 1 and 2093 are divisors of 2093)
2093 / 2 = 1046.5 (the remainder is 1, so 2 is not a divisor of 2093)
2093 / 3 = 697.66666666667 (the remainder is 2, so 3 is not a divisor of 2093)
...
2093 / 44 = 47.568181818182 (the remainder is 25, so 44 is not a divisor of 2093)
2093 / 45 = 46.511111111111 (the remainder is 23, so 45 is not a divisor of 2093)