What are the divisors of 310?

1, 2, 5, 10, 31, 62, 155, 310

There is a total of 8 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 72.

4 even divisors

2, 10, 62, 310

4 odd divisors

1, 5, 31, 155

How to compute the divisors of 310?

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 310 by each of the numbers from 1 to 310 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)

310 / 1 = 310 (the remainder is 0, so 1 is a divisor of 310)
310 / 2 = 155 (the remainder is 0, so 2 is a divisor of 310)
310 / 3 = 103.33333333333 (the remainder is 1, so 3 is not a divisor of 310)
...
310 / 309 = 1.0032362459547 (the remainder is 1, so 309 is not a divisor of 310)
310 / 310 = 1 (the remainder is 0, so 310 is a divisor of 310)

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 310 (i.e. 17.606816861659). 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 $\frac{N}{d} = D$ )

310 / 1 = 310 (the remainder is 0, so 1 and 310 are divisors of 310)
310 / 2 = 155 (the remainder is 0, so 2 and 155 are divisors of 310)
310 / 3 = 103.33333333333 (the remainder is 1, so 3 is not a divisor of 310)
...
310 / 16 = 19.375 (the remainder is 6, so 16 is not a divisor of 310)
310 / 17 = 18.235294117647 (the remainder is 4, so 17 is not a divisor of 310)