What are the divisors of 314?

1, 2, 157, 314

There is a total of 4 positive divisors.
The sum of these divisors is 474.
The arithmetic mean is 118.5.

2 even divisors

2, 314

2 odd divisors

1, 157

How to compute the divisors of 314?

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 314 by each of the numbers from 1 to 314 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)

314 / 1 = 314 (the remainder is 0, so 1 is a divisor of 314)
314 / 2 = 157 (the remainder is 0, so 2 is a divisor of 314)
314 / 3 = 104.66666666667 (the remainder is 2, so 3 is not a divisor of 314)
...
314 / 313 = 1.0031948881789 (the remainder is 1, so 313 is not a divisor of 314)
314 / 314 = 1 (the remainder is 0, so 314 is a divisor of 314)

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 314 (i.e. 17.720045146669). 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 $\frac{N}{d} = D$ )

314 / 1 = 314 (the remainder is 0, so 1 and 314 are divisors of 314)
314 / 2 = 157 (the remainder is 0, so 2 and 157 are divisors of 314)
314 / 3 = 104.66666666667 (the remainder is 2, so 3 is not a divisor of 314)
...
314 / 16 = 19.625 (the remainder is 10, so 16 is not a divisor of 314)
314 / 17 = 18.470588235294 (the remainder is 8, so 17 is not a divisor of 314)