What are the divisors of 3514?

1, 2, 7, 14, 251, 502, 1757, 3514

There is a total of 8 positive divisors.
The sum of these divisors is 6048.
The arithmetic mean is 756.

4 even divisors

2, 14, 502, 3514

4 odd divisors

1, 7, 251, 1757

How to compute the divisors of 3514?

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 3514 by each of the numbers from 1 to 3514 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)

3514 / 1 = 3514 (the remainder is 0, so 1 is a divisor of 3514)
3514 / 2 = 1757 (the remainder is 0, so 2 is a divisor of 3514)
3514 / 3 = 1171.3333333333 (the remainder is 1, so 3 is not a divisor of 3514)
...
3514 / 3513 = 1.0002846569883 (the remainder is 1, so 3513 is not a divisor of 3514)
3514 / 3514 = 1 (the remainder is 0, so 3514 is a divisor of 3514)

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 3514 (i.e. 59.279001341116). 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 $\frac{N}{d} = D$ )

3514 / 1 = 3514 (the remainder is 0, so 1 and 3514 are divisors of 3514)
3514 / 2 = 1757 (the remainder is 0, so 2 and 1757 are divisors of 3514)
3514 / 3 = 1171.3333333333 (the remainder is 1, so 3 is not a divisor of 3514)
...
3514 / 58 = 60.586206896552 (the remainder is 34, so 58 is not a divisor of 3514)
3514 / 59 = 59.559322033898 (the remainder is 33, so 59 is not a divisor of 3514)