What are the divisors of 7007?

1, 7, 11, 13, 49, 77, 91, 143, 539, 637, 1001, 7007

There is a total of 12 positive divisors.
The sum of these divisors is 9576.
The arithmetic mean is 798.

12 odd divisors

1, 7, 11, 13, 49, 77, 91, 143, 539, 637, 1001, 7007

How to compute the divisors of 7007?

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 7007 by each of the numbers from 1 to 7007 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)

7007 / 1 = 7007 (the remainder is 0, so 1 is a divisor of 7007)
7007 / 2 = 3503.5 (the remainder is 1, so 2 is not a divisor of 7007)
7007 / 3 = 2335.6666666667 (the remainder is 2, so 3 is not a divisor of 7007)
...
7007 / 7006 = 1.0001427347987 (the remainder is 1, so 7006 is not a divisor of 7007)
7007 / 7007 = 1 (the remainder is 0, so 7007 is a divisor of 7007)

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 7007 (i.e. 83.70782520171). 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 $\frac{N}{d} = D$ )

7007 / 1 = 7007 (the remainder is 0, so 1 and 7007 are divisors of 7007)
7007 / 2 = 3503.5 (the remainder is 1, so 2 is not a divisor of 7007)
7007 / 3 = 2335.6666666667 (the remainder is 2, so 3 is not a divisor of 7007)
...
7007 / 82 = 85.451219512195 (the remainder is 37, so 82 is not a divisor of 7007)
7007 / 83 = 84.421686746988 (the remainder is 35, so 83 is not a divisor of 7007)