What are the divisors of 2047?

1, 23, 89, 2047

There is a total of 4 positive divisors.
The sum of these divisors is 2160.
The arithmetic mean is 540.

4 odd divisors

1, 23, 89, 2047

How to compute the divisors of 2047?

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 2047 by each of the numbers from 1 to 2047 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)

2047 / 1 = 2047 (the remainder is 0, so 1 is a divisor of 2047)
2047 / 2 = 1023.5 (the remainder is 1, so 2 is not a divisor of 2047)
2047 / 3 = 682.33333333333 (the remainder is 1, so 3 is not a divisor of 2047)
...
2047 / 2046 = 1.0004887585533 (the remainder is 1, so 2046 is not a divisor of 2047)
2047 / 2047 = 1 (the remainder is 0, so 2047 is a divisor of 2047)

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 2047 (i.e. 45.243784103454). 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 $\frac{N}{d} = D$ )

2047 / 1 = 2047 (the remainder is 0, so 1 and 2047 are divisors of 2047)
2047 / 2 = 1023.5 (the remainder is 1, so 2 is not a divisor of 2047)
2047 / 3 = 682.33333333333 (the remainder is 1, so 3 is not a divisor of 2047)
...
2047 / 44 = 46.522727272727 (the remainder is 23, so 44 is not a divisor of 2047)
2047 / 45 = 45.488888888889 (the remainder is 22, so 45 is not a divisor of 2047)