What are the divisors of 514?

1, 2, 257, 514

There is a total of 4 positive divisors.
The sum of these divisors is 774.
The arithmetic mean is 193.5.

2 even divisors

2, 514

2 odd divisors

1, 257

How to compute the divisors of 514?

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 514 by each of the numbers from 1 to 514 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)

514 / 1 = 514 (the remainder is 0, so 1 is a divisor of 514)
514 / 2 = 257 (the remainder is 0, so 2 is a divisor of 514)
514 / 3 = 171.33333333333 (the remainder is 1, so 3 is not a divisor of 514)
...
514 / 513 = 1.0019493177388 (the remainder is 1, so 513 is not a divisor of 514)
514 / 514 = 1 (the remainder is 0, so 514 is a divisor of 514)

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 514 (i.e. 22.671568097509). 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 $\frac{N}{d} = D$ )

514 / 1 = 514 (the remainder is 0, so 1 and 514 are divisors of 514)
514 / 2 = 257 (the remainder is 0, so 2 and 257 are divisors of 514)
514 / 3 = 171.33333333333 (the remainder is 1, so 3 is not a divisor of 514)
...
514 / 21 = 24.47619047619 (the remainder is 10, so 21 is not a divisor of 514)
514 / 22 = 23.363636363636 (the remainder is 8, so 22 is not a divisor of 514)