What are the divisors of 227?

1, 227

There is a total of 2 positive divisors.
The sum of these divisors is 228.
The arithmetic mean is 114.

2 odd divisors

1, 227

How to compute the divisors of 227?

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 227 by each of the numbers from 1 to 227 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)

227 / 1 = 227 (the remainder is 0, so 1 is a divisor of 227)
227 / 2 = 113.5 (the remainder is 1, so 2 is not a divisor of 227)
227 / 3 = 75.666666666667 (the remainder is 2, so 3 is not a divisor of 227)
...
227 / 226 = 1.0044247787611 (the remainder is 1, so 226 is not a divisor of 227)
227 / 227 = 1 (the remainder is 0, so 227 is a divisor of 227)

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 227 (i.e. 15.066519173319). 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 $\frac{N}{d} = D$ )

227 / 1 = 227 (the remainder is 0, so 1 and 227 are divisors of 227)
227 / 2 = 113.5 (the remainder is 1, so 2 is not a divisor of 227)
227 / 3 = 75.666666666667 (the remainder is 2, so 3 is not a divisor of 227)
...
227 / 14 = 16.214285714286 (the remainder is 3, so 14 is not a divisor of 227)
227 / 15 = 15.133333333333 (the remainder is 2, so 15 is not a divisor of 227)