What are the divisors of 257?

1, 257

There is a total of 2 positive divisors.
The sum of these divisors is 258.
The arithmetic mean is 129.

2 odd divisors

1, 257

How to compute the divisors of 257?

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 257 by each of the numbers from 1 to 257 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)

257 / 1 = 257 (the remainder is 0, so 1 is a divisor of 257)
257 / 2 = 128.5 (the remainder is 1, so 2 is not a divisor of 257)
257 / 3 = 85.666666666667 (the remainder is 2, so 3 is not a divisor of 257)
...
257 / 256 = 1.00390625 (the remainder is 1, so 256 is not a divisor of 257)
257 / 257 = 1 (the remainder is 0, so 257 is a divisor of 257)

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 257 (i.e. 16.031219541881). 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 $\frac{N}{d} = D$ )

257 / 1 = 257 (the remainder is 0, so 1 and 257 are divisors of 257)
257 / 2 = 128.5 (the remainder is 1, so 2 is not a divisor of 257)
257 / 3 = 85.666666666667 (the remainder is 2, so 3 is not a divisor of 257)
...
257 / 15 = 17.133333333333 (the remainder is 2, so 15 is not a divisor of 257)
257 / 16 = 16.0625 (the remainder is 1, so 16 is not a divisor of 257)