What are the divisors of 3407?

1, 3407

There is a total of 2 positive divisors.
The sum of these divisors is 3408.
The arithmetic mean is 1704.

2 odd divisors

1, 3407

How to compute the divisors of 3407?

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 3407 by each of the numbers from 1 to 3407 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)

3407 / 1 = 3407 (the remainder is 0, so 1 is a divisor of 3407)
3407 / 2 = 1703.5 (the remainder is 1, so 2 is not a divisor of 3407)
3407 / 3 = 1135.6666666667 (the remainder is 2, so 3 is not a divisor of 3407)
...
3407 / 3406 = 1.0002935995302 (the remainder is 1, so 3406 is not a divisor of 3407)
3407 / 3407 = 1 (the remainder is 0, so 3407 is a divisor of 3407)

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 3407 (i.e. 58.36951259005). 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 $\frac{N}{d} = D$ )

3407 / 1 = 3407 (the remainder is 0, so 1 and 3407 are divisors of 3407)
3407 / 2 = 1703.5 (the remainder is 1, so 2 is not a divisor of 3407)
3407 / 3 = 1135.6666666667 (the remainder is 2, so 3 is not a divisor of 3407)
...
3407 / 57 = 59.771929824561 (the remainder is 44, so 57 is not a divisor of 3407)
3407 / 58 = 58.741379310345 (the remainder is 43, so 58 is not a divisor of 3407)