What are the divisors of 4993?

1, 4993

There is a total of 2 positive divisors.
The sum of these divisors is 4994.
The arithmetic mean is 2497.

2 odd divisors

1, 4993

How to compute the divisors of 4993?

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 4993 by each of the numbers from 1 to 4993 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)

4993 / 1 = 4993 (the remainder is 0, so 1 is a divisor of 4993)
4993 / 2 = 2496.5 (the remainder is 1, so 2 is not a divisor of 4993)
4993 / 3 = 1664.3333333333 (the remainder is 1, so 3 is not a divisor of 4993)
...
4993 / 4992 = 1.0002003205128 (the remainder is 1, so 4992 is not a divisor of 4993)
4993 / 4993 = 1 (the remainder is 0, so 4993 is a divisor of 4993)

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 4993 (i.e. 70.661163307718). 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 $\frac{N}{d} = D$ )

4993 / 1 = 4993 (the remainder is 0, so 1 and 4993 are divisors of 4993)
4993 / 2 = 2496.5 (the remainder is 1, so 2 is not a divisor of 4993)
4993 / 3 = 1664.3333333333 (the remainder is 1, so 3 is not a divisor of 4993)
...
4993 / 69 = 72.36231884058 (the remainder is 25, so 69 is not a divisor of 4993)
4993 / 70 = 71.328571428571 (the remainder is 23, so 70 is not a divisor of 4993)