What are the divisors of 2219?

1, 7, 317, 2219

There is a total of 4 positive divisors.
The sum of these divisors is 2544.
The arithmetic mean is 636.

4 odd divisors

1, 7, 317, 2219

How to compute the divisors of 2219?

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 2219 by each of the numbers from 1 to 2219 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)

2219 / 1 = 2219 (the remainder is 0, so 1 is a divisor of 2219)
2219 / 2 = 1109.5 (the remainder is 1, so 2 is not a divisor of 2219)
2219 / 3 = 739.66666666667 (the remainder is 2, so 3 is not a divisor of 2219)
...
2219 / 2218 = 1.0004508566276 (the remainder is 1, so 2218 is not a divisor of 2219)
2219 / 2219 = 1 (the remainder is 0, so 2219 is a divisor of 2219)

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 2219 (i.e. 47.106262853256). 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 $\frac{N}{d} = D$ )

2219 / 1 = 2219 (the remainder is 0, so 1 and 2219 are divisors of 2219)
2219 / 2 = 1109.5 (the remainder is 1, so 2 is not a divisor of 2219)
2219 / 3 = 739.66666666667 (the remainder is 2, so 3 is not a divisor of 2219)
...
2219 / 46 = 48.239130434783 (the remainder is 11, so 46 is not a divisor of 2219)
2219 / 47 = 47.212765957447 (the remainder is 10, so 47 is not a divisor of 2219)