What are the divisors of 305?

1, 5, 61, 305

There is a total of 4 positive divisors.
The sum of these divisors is 372.
The arithmetic mean is 93.

4 odd divisors

1, 5, 61, 305

How to compute the divisors of 305?

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 305 by each of the numbers from 1 to 305 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)

305 / 1 = 305 (the remainder is 0, so 1 is a divisor of 305)
305 / 2 = 152.5 (the remainder is 1, so 2 is not a divisor of 305)
305 / 3 = 101.66666666667 (the remainder is 2, so 3 is not a divisor of 305)
...
305 / 304 = 1.0032894736842 (the remainder is 1, so 304 is not a divisor of 305)
305 / 305 = 1 (the remainder is 0, so 305 is a divisor of 305)

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 305 (i.e. 17.464249196573). 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 $\frac{N}{d} = D$ )

305 / 1 = 305 (the remainder is 0, so 1 and 305 are divisors of 305)
305 / 2 = 152.5 (the remainder is 1, so 2 is not a divisor of 305)
305 / 3 = 101.66666666667 (the remainder is 2, so 3 is not a divisor of 305)
...
305 / 16 = 19.0625 (the remainder is 1, so 16 is not a divisor of 305)
305 / 17 = 17.941176470588 (the remainder is 16, so 17 is not a divisor of 305)