What are the divisors of 250?

1, 2, 5, 10, 25, 50, 125, 250

There is a total of 8 positive divisors.
The sum of these divisors is 468.
The arithmetic mean is 58.5.

4 even divisors

2, 10, 50, 250

4 odd divisors

1, 5, 25, 125

How to compute the divisors of 250?

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 250 by each of the numbers from 1 to 250 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)

250 / 1 = 250 (the remainder is 0, so 1 is a divisor of 250)
250 / 2 = 125 (the remainder is 0, so 2 is a divisor of 250)
250 / 3 = 83.333333333333 (the remainder is 1, so 3 is not a divisor of 250)
...
250 / 249 = 1.004016064257 (the remainder is 1, so 249 is not a divisor of 250)
250 / 250 = 1 (the remainder is 0, so 250 is a divisor of 250)

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 250 (i.e. 15.811388300842). 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 $\frac{N}{d} = D$ )

250 / 1 = 250 (the remainder is 0, so 1 and 250 are divisors of 250)
250 / 2 = 125 (the remainder is 0, so 2 and 125 are divisors of 250)
250 / 3 = 83.333333333333 (the remainder is 1, so 3 is not a divisor of 250)
...
250 / 14 = 17.857142857143 (the remainder is 12, so 14 is not a divisor of 250)
250 / 15 = 16.666666666667 (the remainder is 10, so 15 is not a divisor of 250)