What are the divisors of 269?

1, 269

There is a total of 2 positive divisors.
The sum of these divisors is 270.
The arithmetic mean is 135.

2 odd divisors

1, 269

How to compute the divisors of 269?

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 269 by each of the numbers from 1 to 269 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)

269 / 1 = 269 (the remainder is 0, so 1 is a divisor of 269)
269 / 2 = 134.5 (the remainder is 1, so 2 is not a divisor of 269)
269 / 3 = 89.666666666667 (the remainder is 2, so 3 is not a divisor of 269)
...
269 / 268 = 1.0037313432836 (the remainder is 1, so 268 is not a divisor of 269)
269 / 269 = 1 (the remainder is 0, so 269 is a divisor of 269)

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 269 (i.e. 16.401219466857). 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 $\frac{N}{d} = D$ )

269 / 1 = 269 (the remainder is 0, so 1 and 269 are divisors of 269)
269 / 2 = 134.5 (the remainder is 1, so 2 is not a divisor of 269)
269 / 3 = 89.666666666667 (the remainder is 2, so 3 is not a divisor of 269)
...
269 / 15 = 17.933333333333 (the remainder is 14, so 15 is not a divisor of 269)
269 / 16 = 16.8125 (the remainder is 13, so 16 is not a divisor of 269)