What are the divisors of 289?

1, 17, 289

There is a total of 3 positive divisors.
The sum of these divisors is 307.
The arithmetic mean is 102.33333333333.

3 odd divisors

1, 17, 289

How to compute the divisors of 289?

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 289 by each of the numbers from 1 to 289 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)

289 / 1 = 289 (the remainder is 0, so 1 is a divisor of 289)
289 / 2 = 144.5 (the remainder is 1, so 2 is not a divisor of 289)
289 / 3 = 96.333333333333 (the remainder is 1, so 3 is not a divisor of 289)
...
289 / 288 = 1.0034722222222 (the remainder is 1, so 288 is not a divisor of 289)
289 / 289 = 1 (the remainder is 0, so 289 is a divisor of 289)

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 289 (i.e. 17). 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 $\frac{N}{d} = D$ )

289 / 1 = 289 (the remainder is 0, so 1 and 289 are divisors of 289)
289 / 2 = 144.5 (the remainder is 1, so 2 is not a divisor of 289)
289 / 3 = 96.333333333333 (the remainder is 1, so 3 is not a divisor of 289)
...
289 / 16 = 18.0625 (the remainder is 1, so 16 is not a divisor of 289)
289 / 17 = 17 (the remainder is 0, so 17 and 17 are divisors of 289)