What are the divisors of 9643?

1, 9643

There is a total of 2 positive divisors.
The sum of these divisors is 9644.
The arithmetic mean is 4822.

2 odd divisors

1, 9643

How to compute the divisors of 9643?

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 9643 by each of the numbers from 1 to 9643 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)

9643 / 1 = 9643 (the remainder is 0, so 1 is a divisor of 9643)
9643 / 2 = 4821.5 (the remainder is 1, so 2 is not a divisor of 9643)
9643 / 3 = 3214.3333333333 (the remainder is 1, so 3 is not a divisor of 9643)
...
9643 / 9642 = 1.0001037129226 (the remainder is 1, so 9642 is not a divisor of 9643)
9643 / 9643 = 1 (the remainder is 0, so 9643 is a divisor of 9643)

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 9643 (i.e. 98.19877799647). 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 $\frac{N}{d} = D$ )

9643 / 1 = 9643 (the remainder is 0, so 1 and 9643 are divisors of 9643)
9643 / 2 = 4821.5 (the remainder is 1, so 2 is not a divisor of 9643)
9643 / 3 = 3214.3333333333 (the remainder is 1, so 3 is not a divisor of 9643)
...
9643 / 97 = 99.412371134021 (the remainder is 40, so 97 is not a divisor of 9643)
9643 / 98 = 98.397959183673 (the remainder is 39, so 98 is not a divisor of 9643)