What are the divisors of 3414?

1, 2, 3, 6, 569, 1138, 1707, 3414

There is a total of 8 positive divisors.
The sum of these divisors is 6840.
The arithmetic mean is 855.

4 even divisors

2, 6, 1138, 3414

4 odd divisors

1, 3, 569, 1707

How to compute the divisors of 3414?

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 3414 by each of the numbers from 1 to 3414 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)

3414 / 1 = 3414 (the remainder is 0, so 1 is a divisor of 3414)
3414 / 2 = 1707 (the remainder is 0, so 2 is a divisor of 3414)
3414 / 3 = 1138 (the remainder is 0, so 3 is a divisor of 3414)
...
3414 / 3413 = 1.000292997363 (the remainder is 1, so 3413 is not a divisor of 3414)
3414 / 3414 = 1 (the remainder is 0, so 3414 is a divisor of 3414)

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 3414 (i.e. 58.429444631966). 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 $\frac{N}{d} = D$ )

3414 / 1 = 3414 (the remainder is 0, so 1 and 3414 are divisors of 3414)
3414 / 2 = 1707 (the remainder is 0, so 2 and 1707 are divisors of 3414)
3414 / 3 = 1138 (the remainder is 0, so 3 and 1138 are divisors of 3414)
...
3414 / 57 = 59.894736842105 (the remainder is 51, so 57 is not a divisor of 3414)
3414 / 58 = 58.862068965517 (the remainder is 50, so 58 is not a divisor of 3414)