What are the divisors of 1635?

1, 3, 5, 15, 109, 327, 545, 1635

There is a total of 8 positive divisors.
The sum of these divisors is 2640.
The arithmetic mean is 330.

8 odd divisors

1, 3, 5, 15, 109, 327, 545, 1635

How to compute the divisors of 1635?

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 1635 by each of the numbers from 1 to 1635 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)

1635 / 1 = 1635 (the remainder is 0, so 1 is a divisor of 1635)
1635 / 2 = 817.5 (the remainder is 1, so 2 is not a divisor of 1635)
1635 / 3 = 545 (the remainder is 0, so 3 is a divisor of 1635)
...
1635 / 1634 = 1.000611995104 (the remainder is 1, so 1634 is not a divisor of 1635)
1635 / 1635 = 1 (the remainder is 0, so 1635 is a divisor of 1635)

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 1635 (i.e. 40.435133238311). 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 $\frac{N}{d} = D$ )

1635 / 1 = 1635 (the remainder is 0, so 1 and 1635 are divisors of 1635)
1635 / 2 = 817.5 (the remainder is 1, so 2 is not a divisor of 1635)
1635 / 3 = 545 (the remainder is 0, so 3 and 545 are divisors of 1635)
...
1635 / 39 = 41.923076923077 (the remainder is 36, so 39 is not a divisor of 1635)
1635 / 40 = 40.875 (the remainder is 35, so 40 is not a divisor of 1635)