What are the divisors of 4093?

1, 4093

There is a total of 2 positive divisors.
The sum of these divisors is 4094.
The arithmetic mean is 2047.

2 odd divisors

1, 4093

How to compute the divisors of 4093?

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 4093 by each of the numbers from 1 to 4093 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)

4093 / 1 = 4093 (the remainder is 0, so 1 is a divisor of 4093)
4093 / 2 = 2046.5 (the remainder is 1, so 2 is not a divisor of 4093)
4093 / 3 = 1364.3333333333 (the remainder is 1, so 3 is not a divisor of 4093)
...
4093 / 4092 = 1.0002443792766 (the remainder is 1, so 4092 is not a divisor of 4093)
4093 / 4093 = 1 (the remainder is 0, so 4093 is a divisor of 4093)

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 4093 (i.e. 63.976558206893). 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 $\frac{N}{d} = D$ )

4093 / 1 = 4093 (the remainder is 0, so 1 and 4093 are divisors of 4093)
4093 / 2 = 2046.5 (the remainder is 1, so 2 is not a divisor of 4093)
4093 / 3 = 1364.3333333333 (the remainder is 1, so 3 is not a divisor of 4093)
...
4093 / 62 = 66.016129032258 (the remainder is 1, so 62 is not a divisor of 4093)
4093 / 63 = 64.968253968254 (the remainder is 61, so 63 is not a divisor of 4093)