What are the divisors of 4219?

1, 4219

There is a total of 2 positive divisors.
The sum of these divisors is 4220.
The arithmetic mean is 2110.

2 odd divisors

1, 4219

How to compute the divisors of 4219?

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 4219 by each of the numbers from 1 to 4219 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)

4219 / 1 = 4219 (the remainder is 0, so 1 is a divisor of 4219)
4219 / 2 = 2109.5 (the remainder is 1, so 2 is not a divisor of 4219)
4219 / 3 = 1406.3333333333 (the remainder is 1, so 3 is not a divisor of 4219)
...
4219 / 4218 = 1.0002370791844 (the remainder is 1, so 4218 is not a divisor of 4219)
4219 / 4219 = 1 (the remainder is 0, so 4219 is a divisor of 4219)

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 4219 (i.e. 64.95382975622). 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 $\frac{N}{d} = D$ )

4219 / 1 = 4219 (the remainder is 0, so 1 and 4219 are divisors of 4219)
4219 / 2 = 2109.5 (the remainder is 1, so 2 is not a divisor of 4219)
4219 / 3 = 1406.3333333333 (the remainder is 1, so 3 is not a divisor of 4219)
...
4219 / 63 = 66.968253968254 (the remainder is 61, so 63 is not a divisor of 4219)
4219 / 64 = 65.921875 (the remainder is 59, so 64 is not a divisor of 4219)