What are the divisors of 9001?

1, 9001

There is a total of 2 positive divisors.
The sum of these divisors is 9002.
The arithmetic mean is 4501.

2 odd divisors

1, 9001

How to compute the divisors of 9001?

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 9001 by each of the numbers from 1 to 9001 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)

9001 / 1 = 9001 (the remainder is 0, so 1 is a divisor of 9001)
9001 / 2 = 4500.5 (the remainder is 1, so 2 is not a divisor of 9001)
9001 / 3 = 3000.3333333333 (the remainder is 1, so 3 is not a divisor of 9001)
...
9001 / 9000 = 1.0001111111111 (the remainder is 1, so 9000 is not a divisor of 9001)
9001 / 9001 = 1 (the remainder is 0, so 9001 is a divisor of 9001)

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 9001 (i.e. 94.873600121425). 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 $\frac{N}{d} = D$ )

9001 / 1 = 9001 (the remainder is 0, so 1 and 9001 are divisors of 9001)
9001 / 2 = 4500.5 (the remainder is 1, so 2 is not a divisor of 9001)
9001 / 3 = 3000.3333333333 (the remainder is 1, so 3 is not a divisor of 9001)
...
9001 / 93 = 96.784946236559 (the remainder is 73, so 93 is not a divisor of 9001)
9001 / 94 = 95.755319148936 (the remainder is 71, so 94 is not a divisor of 9001)