What are the divisors of 1514?

1, 2, 757, 1514

There is a total of 4 positive divisors.
The sum of these divisors is 2274.
The arithmetic mean is 568.5.

2 even divisors

2, 1514

2 odd divisors

1, 757

How to compute the divisors of 1514?

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 1514 by each of the numbers from 1 to 1514 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)

1514 / 1 = 1514 (the remainder is 0, so 1 is a divisor of 1514)
1514 / 2 = 757 (the remainder is 0, so 2 is a divisor of 1514)
1514 / 3 = 504.66666666667 (the remainder is 2, so 3 is not a divisor of 1514)
...
1514 / 1513 = 1.0006609385327 (the remainder is 1, so 1513 is not a divisor of 1514)
1514 / 1514 = 1 (the remainder is 0, so 1514 is a divisor of 1514)

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 1514 (i.e. 38.910152916687). 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 $\frac{N}{d} = D$ )

1514 / 1 = 1514 (the remainder is 0, so 1 and 1514 are divisors of 1514)
1514 / 2 = 757 (the remainder is 0, so 2 and 757 are divisors of 1514)
1514 / 3 = 504.66666666667 (the remainder is 2, so 3 is not a divisor of 1514)
...
1514 / 37 = 40.918918918919 (the remainder is 34, so 37 is not a divisor of 1514)
1514 / 38 = 39.842105263158 (the remainder is 32, so 38 is not a divisor of 1514)