What are the divisors of 233?

1, 233

There is a total of 2 positive divisors.
The sum of these divisors is 234.
The arithmetic mean is 117.

2 odd divisors

1, 233

How to compute the divisors of 233?

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 233 by each of the numbers from 1 to 233 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)

233 / 1 = 233 (the remainder is 0, so 1 is a divisor of 233)
233 / 2 = 116.5 (the remainder is 1, so 2 is not a divisor of 233)
233 / 3 = 77.666666666667 (the remainder is 2, so 3 is not a divisor of 233)
...
233 / 232 = 1.0043103448276 (the remainder is 1, so 232 is not a divisor of 233)
233 / 233 = 1 (the remainder is 0, so 233 is a divisor of 233)

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 233 (i.e. 15.264337522474). 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 $\frac{N}{d} = D$ )

233 / 1 = 233 (the remainder is 0, so 1 and 233 are divisors of 233)
233 / 2 = 116.5 (the remainder is 1, so 2 is not a divisor of 233)
233 / 3 = 77.666666666667 (the remainder is 2, so 3 is not a divisor of 233)
...
233 / 14 = 16.642857142857 (the remainder is 9, so 14 is not a divisor of 233)
233 / 15 = 15.533333333333 (the remainder is 8, so 15 is not a divisor of 233)