What are the divisors of 268?

1, 2, 4, 67, 134, 268

There is a total of 6 positive divisors.
The sum of these divisors is 476.
The arithmetic mean is 79.333333333333.

4 even divisors

2, 4, 134, 268

2 odd divisors

1, 67

How to compute the divisors of 268?

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 268 by each of the numbers from 1 to 268 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)

268 / 1 = 268 (the remainder is 0, so 1 is a divisor of 268)
268 / 2 = 134 (the remainder is 0, so 2 is a divisor of 268)
268 / 3 = 89.333333333333 (the remainder is 1, so 3 is not a divisor of 268)
...
268 / 267 = 1.0037453183521 (the remainder is 1, so 267 is not a divisor of 268)
268 / 268 = 1 (the remainder is 0, so 268 is a divisor of 268)

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 268 (i.e. 16.370705543745). 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 $\frac{N}{d} = D$ )

268 / 1 = 268 (the remainder is 0, so 1 and 268 are divisors of 268)
268 / 2 = 134 (the remainder is 0, so 2 and 134 are divisors of 268)
268 / 3 = 89.333333333333 (the remainder is 1, so 3 is not a divisor of 268)
...
268 / 15 = 17.866666666667 (the remainder is 13, so 15 is not a divisor of 268)
268 / 16 = 16.75 (the remainder is 12, so 16 is not a divisor of 268)