What are the divisors of 251?

1, 251

There is a total of 2 positive divisors.
The sum of these divisors is 252.
The arithmetic mean is 126.

2 odd divisors

1, 251

How to compute the divisors of 251?

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 251 by each of the numbers from 1 to 251 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)

251 / 1 = 251 (the remainder is 0, so 1 is a divisor of 251)
251 / 2 = 125.5 (the remainder is 1, so 2 is not a divisor of 251)
251 / 3 = 83.666666666667 (the remainder is 2, so 3 is not a divisor of 251)
...
251 / 250 = 1.004 (the remainder is 1, so 250 is not a divisor of 251)
251 / 251 = 1 (the remainder is 0, so 251 is a divisor of 251)

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 251 (i.e. 15.842979517755). 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 $\frac{N}{d} = D$ )

251 / 1 = 251 (the remainder is 0, so 1 and 251 are divisors of 251)
251 / 2 = 125.5 (the remainder is 1, so 2 is not a divisor of 251)
251 / 3 = 83.666666666667 (the remainder is 2, so 3 is not a divisor of 251)
...
251 / 14 = 17.928571428571 (the remainder is 13, so 14 is not a divisor of 251)
251 / 15 = 16.733333333333 (the remainder is 11, so 15 is not a divisor of 251)