What are the divisors of 2137?

1, 2137

There is a total of 2 positive divisors.
The sum of these divisors is 2138.
The arithmetic mean is 1069.

2 odd divisors

1, 2137

How to compute the divisors of 2137?

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 2137 by each of the numbers from 1 to 2137 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)

2137 / 1 = 2137 (the remainder is 0, so 1 is a divisor of 2137)
2137 / 2 = 1068.5 (the remainder is 1, so 2 is not a divisor of 2137)
2137 / 3 = 712.33333333333 (the remainder is 1, so 3 is not a divisor of 2137)
...
2137 / 2136 = 1.000468164794 (the remainder is 1, so 2136 is not a divisor of 2137)
2137 / 2137 = 1 (the remainder is 0, so 2137 is a divisor of 2137)

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 2137 (i.e. 46.227697325305). 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 $\frac{N}{d} = D$ )

2137 / 1 = 2137 (the remainder is 0, so 1 and 2137 are divisors of 2137)
2137 / 2 = 1068.5 (the remainder is 1, so 2 is not a divisor of 2137)
2137 / 3 = 712.33333333333 (the remainder is 1, so 3 is not a divisor of 2137)
...
2137 / 45 = 47.488888888889 (the remainder is 22, so 45 is not a divisor of 2137)
2137 / 46 = 46.45652173913 (the remainder is 21, so 46 is not a divisor of 2137)