What are the divisors of 3495?

1, 3, 5, 15, 233, 699, 1165, 3495

There is a total of 8 positive divisors.
The sum of these divisors is 5616.
The arithmetic mean is 702.

8 odd divisors

1, 3, 5, 15, 233, 699, 1165, 3495

How to compute the divisors of 3495?

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 3495 by each of the numbers from 1 to 3495 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)

3495 / 1 = 3495 (the remainder is 0, so 1 is a divisor of 3495)
3495 / 2 = 1747.5 (the remainder is 1, so 2 is not a divisor of 3495)
3495 / 3 = 1165 (the remainder is 0, so 3 is a divisor of 3495)
...
3495 / 3494 = 1.0002862049227 (the remainder is 1, so 3494 is not a divisor of 3495)
3495 / 3495 = 1 (the remainder is 0, so 3495 is a divisor of 3495)

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 3495 (i.e. 59.11852501543). 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 $\frac{N}{d} = D$ )

3495 / 1 = 3495 (the remainder is 0, so 1 and 3495 are divisors of 3495)
3495 / 2 = 1747.5 (the remainder is 1, so 2 is not a divisor of 3495)
3495 / 3 = 1165 (the remainder is 0, so 3 and 1165 are divisors of 3495)
...
3495 / 58 = 60.258620689655 (the remainder is 15, so 58 is not a divisor of 3495)
3495 / 59 = 59.237288135593 (the remainder is 14, so 59 is not a divisor of 3495)