What are the divisors of 1645?

1, 5, 7, 35, 47, 235, 329, 1645

There is a total of 8 positive divisors.
The sum of these divisors is 2304.
The arithmetic mean is 288.

8 odd divisors

1, 5, 7, 35, 47, 235, 329, 1645

How to compute the divisors of 1645?

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 1645 by each of the numbers from 1 to 1645 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)

1645 / 1 = 1645 (the remainder is 0, so 1 is a divisor of 1645)
1645 / 2 = 822.5 (the remainder is 1, so 2 is not a divisor of 1645)
1645 / 3 = 548.33333333333 (the remainder is 1, so 3 is not a divisor of 1645)
...
1645 / 1644 = 1.0006082725061 (the remainder is 1, so 1644 is not a divisor of 1645)
1645 / 1645 = 1 (the remainder is 0, so 1645 is a divisor of 1645)

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 1645 (i.e. 40.558599581346). 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 $\frac{N}{d} = D$ )

1645 / 1 = 1645 (the remainder is 0, so 1 and 1645 are divisors of 1645)
1645 / 2 = 822.5 (the remainder is 1, so 2 is not a divisor of 1645)
1645 / 3 = 548.33333333333 (the remainder is 1, so 3 is not a divisor of 1645)
...
1645 / 39 = 42.179487179487 (the remainder is 7, so 39 is not a divisor of 1645)
1645 / 40 = 41.125 (the remainder is 5, so 40 is not a divisor of 1645)