What are the divisors of 2119?

1, 13, 163, 2119

There is a total of 4 positive divisors.
The sum of these divisors is 2296.
The arithmetic mean is 574.

4 odd divisors

1, 13, 163, 2119

How to compute the divisors of 2119?

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 2119 by each of the numbers from 1 to 2119 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)

2119 / 1 = 2119 (the remainder is 0, so 1 is a divisor of 2119)
2119 / 2 = 1059.5 (the remainder is 1, so 2 is not a divisor of 2119)
2119 / 3 = 706.33333333333 (the remainder is 1, so 3 is not a divisor of 2119)
...
2119 / 2118 = 1.0004721435316 (the remainder is 1, so 2118 is not a divisor of 2119)
2119 / 2119 = 1 (the remainder is 0, so 2119 is a divisor of 2119)

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 2119 (i.e. 46.032597145936). 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 $\frac{N}{d} = D$ )

2119 / 1 = 2119 (the remainder is 0, so 1 and 2119 are divisors of 2119)
2119 / 2 = 1059.5 (the remainder is 1, so 2 is not a divisor of 2119)
2119 / 3 = 706.33333333333 (the remainder is 1, so 3 is not a divisor of 2119)
...
2119 / 45 = 47.088888888889 (the remainder is 4, so 45 is not a divisor of 2119)
2119 / 46 = 46.065217391304 (the remainder is 3, so 46 is not a divisor of 2119)