What are the divisors of 3413?

1, 3413

There is a total of 2 positive divisors.
The sum of these divisors is 3414.
The arithmetic mean is 1707.

2 odd divisors

1, 3413

How to compute the divisors of 3413?

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 3413 by each of the numbers from 1 to 3413 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)

3413 / 1 = 3413 (the remainder is 0, so 1 is a divisor of 3413)
3413 / 2 = 1706.5 (the remainder is 1, so 2 is not a divisor of 3413)
3413 / 3 = 1137.6666666667 (the remainder is 2, so 3 is not a divisor of 3413)
...
3413 / 3412 = 1.0002930832356 (the remainder is 1, so 3412 is not a divisor of 3413)
3413 / 3413 = 1 (the remainder is 0, so 3413 is a divisor of 3413)

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 3413 (i.e. 58.420886675914). 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 $\frac{N}{d} = D$ )

3413 / 1 = 3413 (the remainder is 0, so 1 and 3413 are divisors of 3413)
3413 / 2 = 1706.5 (the remainder is 1, so 2 is not a divisor of 3413)
3413 / 3 = 1137.6666666667 (the remainder is 2, so 3 is not a divisor of 3413)
...
3413 / 57 = 59.877192982456 (the remainder is 50, so 57 is not a divisor of 3413)
3413 / 58 = 58.844827586207 (the remainder is 49, so 58 is not a divisor of 3413)