What are the divisors of 2029?

1, 2029

There is a total of 2 positive divisors.
The sum of these divisors is 2030.
The arithmetic mean is 1015.

2 odd divisors

1, 2029

How to compute the divisors of 2029?

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 2029 by each of the numbers from 1 to 2029 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)

2029 / 1 = 2029 (the remainder is 0, so 1 is a divisor of 2029)
2029 / 2 = 1014.5 (the remainder is 1, so 2 is not a divisor of 2029)
2029 / 3 = 676.33333333333 (the remainder is 1, so 3 is not a divisor of 2029)
...
2029 / 2028 = 1.0004930966469 (the remainder is 1, so 2028 is not a divisor of 2029)
2029 / 2029 = 1 (the remainder is 0, so 2029 is a divisor of 2029)

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 2029 (i.e. 45.044422518221). 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 $\frac{N}{d} = D$ )

2029 / 1 = 2029 (the remainder is 0, so 1 and 2029 are divisors of 2029)
2029 / 2 = 1014.5 (the remainder is 1, so 2 is not a divisor of 2029)
2029 / 3 = 676.33333333333 (the remainder is 1, so 3 is not a divisor of 2029)
...
2029 / 44 = 46.113636363636 (the remainder is 5, so 44 is not a divisor of 2029)
2029 / 45 = 45.088888888889 (the remainder is 4, so 45 is not a divisor of 2029)