What are the divisors of 1001?

1, 7, 11, 13, 77, 91, 143, 1001

There is a total of 8 positive divisors.
The sum of these divisors is 1344.
The arithmetic mean is 168.

8 odd divisors

1, 7, 11, 13, 77, 91, 143, 1001

How to compute the divisors of 1001?

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 1001 by each of the numbers from 1 to 1001 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)

1001 / 1 = 1001 (the remainder is 0, so 1 is a divisor of 1001)
1001 / 2 = 500.5 (the remainder is 1, so 2 is not a divisor of 1001)
1001 / 3 = 333.66666666667 (the remainder is 2, so 3 is not a divisor of 1001)
...
1001 / 1000 = 1.001 (the remainder is 1, so 1000 is not a divisor of 1001)
1001 / 1001 = 1 (the remainder is 0, so 1001 is a divisor of 1001)

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 1001 (i.e. 31.638584039113). 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 $\frac{N}{d} = D$ )

1001 / 1 = 1001 (the remainder is 0, so 1 and 1001 are divisors of 1001)
1001 / 2 = 500.5 (the remainder is 1, so 2 is not a divisor of 1001)
1001 / 3 = 333.66666666667 (the remainder is 2, so 3 is not a divisor of 1001)
...
1001 / 30 = 33.366666666667 (the remainder is 11, so 30 is not a divisor of 1001)
1001 / 31 = 32.290322580645 (the remainder is 9, so 31 is not a divisor of 1001)