What are the divisors of 219?

1, 3, 73, 219

There is a total of 4 positive divisors.
The sum of these divisors is 296.
The arithmetic mean is 74.

4 odd divisors

1, 3, 73, 219

How to compute the divisors of 219?

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 219 by each of the numbers from 1 to 219 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)

219 / 1 = 219 (the remainder is 0, so 1 is a divisor of 219)
219 / 2 = 109.5 (the remainder is 1, so 2 is not a divisor of 219)
219 / 3 = 73 (the remainder is 0, so 3 is a divisor of 219)
...
219 / 218 = 1.0045871559633 (the remainder is 1, so 218 is not a divisor of 219)
219 / 219 = 1 (the remainder is 0, so 219 is a divisor of 219)

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 219 (i.e. 14.798648586949). 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 $\frac{N}{d} = D$ )

219 / 1 = 219 (the remainder is 0, so 1 and 219 are divisors of 219)
219 / 2 = 109.5 (the remainder is 1, so 2 is not a divisor of 219)
219 / 3 = 73 (the remainder is 0, so 3 and 73 are divisors of 219)
...
219 / 13 = 16.846153846154 (the remainder is 11, so 13 is not a divisor of 219)
219 / 14 = 15.642857142857 (the remainder is 9, so 14 is not a divisor of 219)