What are the divisors of 267?

1, 3, 89, 267

There is a total of 4 positive divisors.
The sum of these divisors is 360.
The arithmetic mean is 90.

4 odd divisors

1, 3, 89, 267

How to compute the divisors of 267?

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 267 by each of the numbers from 1 to 267 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)

267 / 1 = 267 (the remainder is 0, so 1 is a divisor of 267)
267 / 2 = 133.5 (the remainder is 1, so 2 is not a divisor of 267)
267 / 3 = 89 (the remainder is 0, so 3 is a divisor of 267)
...
267 / 266 = 1.0037593984962 (the remainder is 1, so 266 is not a divisor of 267)
267 / 267 = 1 (the remainder is 0, so 267 is a divisor of 267)

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 267 (i.e. 16.340134638368). 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 $\frac{N}{d} = D$ )

267 / 1 = 267 (the remainder is 0, so 1 and 267 are divisors of 267)
267 / 2 = 133.5 (the remainder is 1, so 2 is not a divisor of 267)
267 / 3 = 89 (the remainder is 0, so 3 and 89 are divisors of 267)
...
267 / 15 = 17.8 (the remainder is 12, so 15 is not a divisor of 267)
267 / 16 = 16.6875 (the remainder is 11, so 16 is not a divisor of 267)