What are the divisors of 223?

1, 223

There is a total of 2 positive divisors.
The sum of these divisors is 224.
The arithmetic mean is 112.

2 odd divisors

1, 223

How to compute the divisors of 223?

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 223 by each of the numbers from 1 to 223 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)

223 / 1 = 223 (the remainder is 0, so 1 is a divisor of 223)
223 / 2 = 111.5 (the remainder is 1, so 2 is not a divisor of 223)
223 / 3 = 74.333333333333 (the remainder is 1, so 3 is not a divisor of 223)
...
223 / 222 = 1.0045045045045 (the remainder is 1, so 222 is not a divisor of 223)
223 / 223 = 1 (the remainder is 0, so 223 is a divisor of 223)

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 223 (i.e. 14.933184523068). 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 $\frac{N}{d} = D$ )

223 / 1 = 223 (the remainder is 0, so 1 and 223 are divisors of 223)
223 / 2 = 111.5 (the remainder is 1, so 2 is not a divisor of 223)
223 / 3 = 74.333333333333 (the remainder is 1, so 3 is not a divisor of 223)
...
223 / 13 = 17.153846153846 (the remainder is 2, so 13 is not a divisor of 223)
223 / 14 = 15.928571428571 (the remainder is 13, so 14 is not a divisor of 223)