What are the divisors of 3465?

1, 3, 5, 7, 9, 11, 15, 21, 33, 35, 45, 55, 63, 77, 99, 105, 165, 231, 315, 385, 495, 693, 1155, 3465

There is a total of 24 positive divisors.
The sum of these divisors is 7488.
The arithmetic mean is 312.

24 odd divisors

1, 3, 5, 7, 9, 11, 15, 21, 33, 35, 45, 55, 63, 77, 99, 105, 165, 231, 315, 385, 495, 693, 1155, 3465

How to compute the divisors of 3465?

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 3465 by each of the numbers from 1 to 3465 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)

3465 / 1 = 3465 (the remainder is 0, so 1 is a divisor of 3465)
3465 / 2 = 1732.5 (the remainder is 1, so 2 is not a divisor of 3465)
3465 / 3 = 1155 (the remainder is 0, so 3 is a divisor of 3465)
...
3465 / 3464 = 1.0002886836028 (the remainder is 1, so 3464 is not a divisor of 3465)
3465 / 3465 = 1 (the remainder is 0, so 3465 is a divisor of 3465)

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 3465 (i.e. 58.864250611046). 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 $\frac{N}{d} = D$ )

3465 / 1 = 3465 (the remainder is 0, so 1 and 3465 are divisors of 3465)
3465 / 2 = 1732.5 (the remainder is 1, so 2 is not a divisor of 3465)
3465 / 3 = 1155 (the remainder is 0, so 3 and 1155 are divisors of 3465)
...
3465 / 57 = 60.789473684211 (the remainder is 45, so 57 is not a divisor of 3465)
3465 / 58 = 59.741379310345 (the remainder is 43, so 58 is not a divisor of 3465)