What are the divisors of 231?

1, 3, 7, 11, 21, 33, 77, 231

There is a total of 8 positive divisors.
The sum of these divisors is 384.
The arithmetic mean is 48.

8 odd divisors

1, 3, 7, 11, 21, 33, 77, 231

How to compute the divisors of 231?

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 231 by each of the numbers from 1 to 231 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)

231 / 1 = 231 (the remainder is 0, so 1 is a divisor of 231)
231 / 2 = 115.5 (the remainder is 1, so 2 is not a divisor of 231)
231 / 3 = 77 (the remainder is 0, so 3 is a divisor of 231)
...
231 / 230 = 1.004347826087 (the remainder is 1, so 230 is not a divisor of 231)
231 / 231 = 1 (the remainder is 0, so 231 is a divisor of 231)

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 231 (i.e. 15.198684153571). 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 $\frac{N}{d} = D$ )

231 / 1 = 231 (the remainder is 0, so 1 and 231 are divisors of 231)
231 / 2 = 115.5 (the remainder is 1, so 2 is not a divisor of 231)
231 / 3 = 77 (the remainder is 0, so 3 and 77 are divisors of 231)
...
231 / 14 = 16.5 (the remainder is 7, so 14 is not a divisor of 231)
231 / 15 = 15.4 (the remainder is 6, so 15 is not a divisor of 231)