What are the divisors of 3405?

1, 3, 5, 15, 227, 681, 1135, 3405

There is a total of 8 positive divisors.
The sum of these divisors is 5472.
The arithmetic mean is 684.

8 odd divisors

1, 3, 5, 15, 227, 681, 1135, 3405

How to compute the divisors of 3405?

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 3405 by each of the numbers from 1 to 3405 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)

3405 / 1 = 3405 (the remainder is 0, so 1 is a divisor of 3405)
3405 / 2 = 1702.5 (the remainder is 1, so 2 is not a divisor of 3405)
3405 / 3 = 1135 (the remainder is 0, so 3 is a divisor of 3405)
...
3405 / 3404 = 1.0002937720329 (the remainder is 1, so 3404 is not a divisor of 3405)
3405 / 3405 = 1 (the remainder is 0, so 3405 is a divisor of 3405)

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 3405 (i.e. 58.352377843581). 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 $\frac{N}{d} = D$ )

3405 / 1 = 3405 (the remainder is 0, so 1 and 3405 are divisors of 3405)
3405 / 2 = 1702.5 (the remainder is 1, so 2 is not a divisor of 3405)
3405 / 3 = 1135 (the remainder is 0, so 3 and 1135 are divisors of 3405)
...
3405 / 57 = 59.736842105263 (the remainder is 42, so 57 is not a divisor of 3405)
3405 / 58 = 58.706896551724 (the remainder is 41, so 58 is not a divisor of 3405)