What are the divisors of 254?

1, 2, 127, 254

There is a total of 4 positive divisors.
The sum of these divisors is 384.
The arithmetic mean is 96.

2 even divisors

2, 254

2 odd divisors

1, 127

How to compute the divisors of 254?

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 254 by each of the numbers from 1 to 254 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)

254 / 1 = 254 (the remainder is 0, so 1 is a divisor of 254)
254 / 2 = 127 (the remainder is 0, so 2 is a divisor of 254)
254 / 3 = 84.666666666667 (the remainder is 2, so 3 is not a divisor of 254)
...
254 / 253 = 1.00395256917 (the remainder is 1, so 253 is not a divisor of 254)
254 / 254 = 1 (the remainder is 0, so 254 is a divisor of 254)

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 254 (i.e. 15.937377450509). 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 $\frac{N}{d} = D$ )

254 / 1 = 254 (the remainder is 0, so 1 and 254 are divisors of 254)
254 / 2 = 127 (the remainder is 0, so 2 and 127 are divisors of 254)
254 / 3 = 84.666666666667 (the remainder is 2, so 3 is not a divisor of 254)
...
254 / 14 = 18.142857142857 (the remainder is 2, so 14 is not a divisor of 254)
254 / 15 = 16.933333333333 (the remainder is 14, so 15 is not a divisor of 254)