What are the divisors of 2129?

1, 2129

There is a total of 2 positive divisors.
The sum of these divisors is 2130.
The arithmetic mean is 1065.

2 odd divisors

1, 2129

How to compute the divisors of 2129?

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 2129 by each of the numbers from 1 to 2129 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)

2129 / 1 = 2129 (the remainder is 0, so 1 is a divisor of 2129)
2129 / 2 = 1064.5 (the remainder is 1, so 2 is not a divisor of 2129)
2129 / 3 = 709.66666666667 (the remainder is 2, so 3 is not a divisor of 2129)
...
2129 / 2128 = 1.000469924812 (the remainder is 1, so 2128 is not a divisor of 2129)
2129 / 2129 = 1 (the remainder is 0, so 2129 is a divisor of 2129)

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 2129 (i.e. 46.141087980237). 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 $\frac{N}{d} = D$ )

2129 / 1 = 2129 (the remainder is 0, so 1 and 2129 are divisors of 2129)
2129 / 2 = 1064.5 (the remainder is 1, so 2 is not a divisor of 2129)
2129 / 3 = 709.66666666667 (the remainder is 2, so 3 is not a divisor of 2129)
...
2129 / 45 = 47.311111111111 (the remainder is 14, so 45 is not a divisor of 2129)
2129 / 46 = 46.282608695652 (the remainder is 13, so 46 is not a divisor of 2129)