What are the divisors of 273?

1, 3, 7, 13, 21, 39, 91, 273

There is a total of 8 positive divisors.
The sum of these divisors is 448.
The arithmetic mean is 56.

8 odd divisors

1, 3, 7, 13, 21, 39, 91, 273

How to compute the divisors of 273?

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 273 by each of the numbers from 1 to 273 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)

273 / 1 = 273 (the remainder is 0, so 1 is a divisor of 273)
273 / 2 = 136.5 (the remainder is 1, so 2 is not a divisor of 273)
273 / 3 = 91 (the remainder is 0, so 3 is a divisor of 273)
...
273 / 272 = 1.0036764705882 (the remainder is 1, so 272 is not a divisor of 273)
273 / 273 = 1 (the remainder is 0, so 273 is a divisor of 273)

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 273 (i.e. 16.522711641858). 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 $\frac{N}{d} = D$ )

273 / 1 = 273 (the remainder is 0, so 1 and 273 are divisors of 273)
273 / 2 = 136.5 (the remainder is 1, so 2 is not a divisor of 273)
273 / 3 = 91 (the remainder is 0, so 3 and 91 are divisors of 273)
...
273 / 15 = 18.2 (the remainder is 3, so 15 is not a divisor of 273)
273 / 16 = 17.0625 (the remainder is 1, so 16 is not a divisor of 273)