What are the divisors of 2167?

1, 11, 197, 2167

There is a total of 4 positive divisors.
The sum of these divisors is 2376.
The arithmetic mean is 594.

4 odd divisors

1, 11, 197, 2167

How to compute the divisors of 2167?

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 2167 by each of the numbers from 1 to 2167 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)

2167 / 1 = 2167 (the remainder is 0, so 1 is a divisor of 2167)
2167 / 2 = 1083.5 (the remainder is 1, so 2 is not a divisor of 2167)
2167 / 3 = 722.33333333333 (the remainder is 1, so 3 is not a divisor of 2167)
...
2167 / 2166 = 1.0004616805171 (the remainder is 1, so 2166 is not a divisor of 2167)
2167 / 2167 = 1 (the remainder is 0, so 2167 is a divisor of 2167)

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 2167 (i.e. 46.55104724923). 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 $\frac{N}{d} = D$ )

2167 / 1 = 2167 (the remainder is 0, so 1 and 2167 are divisors of 2167)
2167 / 2 = 1083.5 (the remainder is 1, so 2 is not a divisor of 2167)
2167 / 3 = 722.33333333333 (the remainder is 1, so 3 is not a divisor of 2167)
...
2167 / 45 = 48.155555555556 (the remainder is 7, so 45 is not a divisor of 2167)
2167 / 46 = 47.108695652174 (the remainder is 5, so 46 is not a divisor of 2167)