What are the divisors of 7000?

1, 2, 4, 5, 7, 8, 10, 14, 20, 25, 28, 35, 40, 50, 56, 70, 100, 125, 140, 175, 200, 250, 280, 350, 500, 700, 875, 1000, 1400, 1750, 3500, 7000

There is a total of 32 positive divisors.
The sum of these divisors is 18720.
The arithmetic mean is 585.

24 even divisors

2, 4, 8, 10, 14, 20, 28, 40, 50, 56, 70, 100, 140, 200, 250, 280, 350, 500, 700, 1000, 1400, 1750, 3500, 7000

8 odd divisors

1, 5, 7, 25, 35, 125, 175, 875

How to compute the divisors of 7000?

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 7000 by each of the numbers from 1 to 7000 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)

7000 / 1 = 7000 (the remainder is 0, so 1 is a divisor of 7000)
7000 / 2 = 3500 (the remainder is 0, so 2 is a divisor of 7000)
7000 / 3 = 2333.3333333333 (the remainder is 1, so 3 is not a divisor of 7000)
...
7000 / 6999 = 1.0001428775539 (the remainder is 1, so 6999 is not a divisor of 7000)
7000 / 7000 = 1 (the remainder is 0, so 7000 is a divisor of 7000)

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 7000 (i.e. 83.666002653408). 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 $\frac{N}{d} = D$ )

7000 / 1 = 7000 (the remainder is 0, so 1 and 7000 are divisors of 7000)
7000 / 2 = 3500 (the remainder is 0, so 2 and 3500 are divisors of 7000)
7000 / 3 = 2333.3333333333 (the remainder is 1, so 3 is not a divisor of 7000)
...
7000 / 82 = 85.365853658537 (the remainder is 30, so 82 is not a divisor of 7000)
7000 / 83 = 84.33734939759 (the remainder is 28, so 83 is not a divisor of 7000)