What are the divisors of 4123?

1, 7, 19, 31, 133, 217, 589, 4123

There is a total of 8 positive divisors.
The sum of these divisors is 5120.
The arithmetic mean is 640.

8 odd divisors

1, 7, 19, 31, 133, 217, 589, 4123

How to compute the divisors of 4123?

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 4123 by each of the numbers from 1 to 4123 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)

4123 / 1 = 4123 (the remainder is 0, so 1 is a divisor of 4123)
4123 / 2 = 2061.5 (the remainder is 1, so 2 is not a divisor of 4123)
4123 / 3 = 1374.3333333333 (the remainder is 1, so 3 is not a divisor of 4123)
...
4123 / 4122 = 1.0002426006793 (the remainder is 1, so 4122 is not a divisor of 4123)
4123 / 4123 = 1 (the remainder is 0, so 4123 is a divisor of 4123)

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 4123 (i.e. 64.210591026715). 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 $\frac{N}{d} = D$ )

4123 / 1 = 4123 (the remainder is 0, so 1 and 4123 are divisors of 4123)
4123 / 2 = 2061.5 (the remainder is 1, so 2 is not a divisor of 4123)
4123 / 3 = 1374.3333333333 (the remainder is 1, so 3 is not a divisor of 4123)
...
4123 / 63 = 65.444444444444 (the remainder is 28, so 63 is not a divisor of 4123)
4123 / 64 = 64.421875 (the remainder is 27, so 64 is not a divisor of 4123)