What are the divisors of 301?

1, 7, 43, 301

There is a total of 4 positive divisors.
The sum of these divisors is 352.
The arithmetic mean is 88.

4 odd divisors

1, 7, 43, 301

How to compute the divisors of 301?

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 301 by each of the numbers from 1 to 301 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)

301 / 1 = 301 (the remainder is 0, so 1 is a divisor of 301)
301 / 2 = 150.5 (the remainder is 1, so 2 is not a divisor of 301)
301 / 3 = 100.33333333333 (the remainder is 1, so 3 is not a divisor of 301)
...
301 / 300 = 1.0033333333333 (the remainder is 1, so 300 is not a divisor of 301)
301 / 301 = 1 (the remainder is 0, so 301 is a divisor of 301)

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 301 (i.e. 17.349351572897). 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 $\frac{N}{d} = D$ )

301 / 1 = 301 (the remainder is 0, so 1 and 301 are divisors of 301)
301 / 2 = 150.5 (the remainder is 1, so 2 is not a divisor of 301)
301 / 3 = 100.33333333333 (the remainder is 1, so 3 is not a divisor of 301)
...
301 / 16 = 18.8125 (the remainder is 13, so 16 is not a divisor of 301)
301 / 17 = 17.705882352941 (the remainder is 12, so 17 is not a divisor of 301)