What are the divisors of 3001?

1, 3001

There is a total of 2 positive divisors.
The sum of these divisors is 3002.
The arithmetic mean is 1501.

2 odd divisors

1, 3001

How to compute the divisors of 3001?

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 3001 by each of the numbers from 1 to 3001 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)

3001 / 1 = 3001 (the remainder is 0, so 1 is a divisor of 3001)
3001 / 2 = 1500.5 (the remainder is 1, so 2 is not a divisor of 3001)
3001 / 3 = 1000.3333333333 (the remainder is 1, so 3 is not a divisor of 3001)
...
3001 / 3000 = 1.0003333333333 (the remainder is 1, so 3000 is not a divisor of 3001)
3001 / 3001 = 1 (the remainder is 0, so 3001 is a divisor of 3001)

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 3001 (i.e. 54.781383699209). 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 $\frac{N}{d} = D$ )

3001 / 1 = 3001 (the remainder is 0, so 1 and 3001 are divisors of 3001)
3001 / 2 = 1500.5 (the remainder is 1, so 2 is not a divisor of 3001)
3001 / 3 = 1000.3333333333 (the remainder is 1, so 3 is not a divisor of 3001)
...
3001 / 53 = 56.622641509434 (the remainder is 33, so 53 is not a divisor of 3001)
3001 / 54 = 55.574074074074 (the remainder is 31, so 54 is not a divisor of 3001)