What are the divisors of 2069?

1, 2069

There is a total of 2 positive divisors.
The sum of these divisors is 2070.
The arithmetic mean is 1035.

2 odd divisors

1, 2069

How to compute the divisors of 2069?

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 2069 by each of the numbers from 1 to 2069 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)

2069 / 1 = 2069 (the remainder is 0, so 1 is a divisor of 2069)
2069 / 2 = 1034.5 (the remainder is 1, so 2 is not a divisor of 2069)
2069 / 3 = 689.66666666667 (the remainder is 2, so 3 is not a divisor of 2069)
...
2069 / 2068 = 1.0004835589942 (the remainder is 1, so 2068 is not a divisor of 2069)
2069 / 2069 = 1 (the remainder is 0, so 2069 is a divisor of 2069)

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 2069 (i.e. 45.486261662177). 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 $\frac{N}{d} = D$ )

2069 / 1 = 2069 (the remainder is 0, so 1 and 2069 are divisors of 2069)
2069 / 2 = 1034.5 (the remainder is 1, so 2 is not a divisor of 2069)
2069 / 3 = 689.66666666667 (the remainder is 2, so 3 is not a divisor of 2069)
...
2069 / 44 = 47.022727272727 (the remainder is 1, so 44 is not a divisor of 2069)
2069 / 45 = 45.977777777778 (the remainder is 44, so 45 is not a divisor of 2069)