What are the divisors of 311?

1, 311

There is a total of 2 positive divisors.
The sum of these divisors is 312.
The arithmetic mean is 156.

2 odd divisors

1, 311

How to compute the divisors of 311?

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 311 by each of the numbers from 1 to 311 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)

311 / 1 = 311 (the remainder is 0, so 1 is a divisor of 311)
311 / 2 = 155.5 (the remainder is 1, so 2 is not a divisor of 311)
311 / 3 = 103.66666666667 (the remainder is 2, so 3 is not a divisor of 311)
...
311 / 310 = 1.0032258064516 (the remainder is 1, so 310 is not a divisor of 311)
311 / 311 = 1 (the remainder is 0, so 311 is a divisor of 311)

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 311 (i.e. 17.635192088548). 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 $\frac{N}{d} = D$ )

311 / 1 = 311 (the remainder is 0, so 1 and 311 are divisors of 311)
311 / 2 = 155.5 (the remainder is 1, so 2 is not a divisor of 311)
311 / 3 = 103.66666666667 (the remainder is 2, so 3 is not a divisor of 311)
...
311 / 16 = 19.4375 (the remainder is 7, so 16 is not a divisor of 311)
311 / 17 = 18.294117647059 (the remainder is 5, so 17 is not a divisor of 311)