What are the divisors of 210?

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210

There is a total of 16 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 36.

8 even divisors

2, 6, 10, 14, 30, 42, 70, 210

8 odd divisors

1, 3, 5, 7, 15, 21, 35, 105

How to compute the divisors of 210?

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 210 by each of the numbers from 1 to 210 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)

210 / 1 = 210 (the remainder is 0, so 1 is a divisor of 210)
210 / 2 = 105 (the remainder is 0, so 2 is a divisor of 210)
210 / 3 = 70 (the remainder is 0, so 3 is a divisor of 210)
...
210 / 209 = 1.0047846889952 (the remainder is 1, so 209 is not a divisor of 210)
210 / 210 = 1 (the remainder is 0, so 210 is a divisor of 210)

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 210 (i.e. 14.491376746189). 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 $\frac{N}{d} = D$ )

210 / 1 = 210 (the remainder is 0, so 1 and 210 are divisors of 210)
210 / 2 = 105 (the remainder is 0, so 2 and 105 are divisors of 210)
210 / 3 = 70 (the remainder is 0, so 3 and 70 are divisors of 210)
...
210 / 13 = 16.153846153846 (the remainder is 2, so 13 is not a divisor of 210)
210 / 14 = 15 (the remainder is 0, so 14 and 15 are divisors of 210)