What are the divisors of 261?

1, 3, 9, 29, 87, 261

There is a total of 6 positive divisors.
The sum of these divisors is 390.
The arithmetic mean is 65.

6 odd divisors

1, 3, 9, 29, 87, 261

How to compute the divisors of 261?

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 261 by each of the numbers from 1 to 261 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)

261 / 1 = 261 (the remainder is 0, so 1 is a divisor of 261)
261 / 2 = 130.5 (the remainder is 1, so 2 is not a divisor of 261)
261 / 3 = 87 (the remainder is 0, so 3 is a divisor of 261)
...
261 / 260 = 1.0038461538462 (the remainder is 1, so 260 is not a divisor of 261)
261 / 261 = 1 (the remainder is 0, so 261 is a divisor of 261)

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 261 (i.e. 16.155494421404). 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 $\frac{N}{d} = D$ )

261 / 1 = 261 (the remainder is 0, so 1 and 261 are divisors of 261)
261 / 2 = 130.5 (the remainder is 1, so 2 is not a divisor of 261)
261 / 3 = 87 (the remainder is 0, so 3 and 87 are divisors of 261)
...
261 / 15 = 17.4 (the remainder is 6, so 15 is not a divisor of 261)
261 / 16 = 16.3125 (the remainder is 5, so 16 is not a divisor of 261)