What are the divisors of 211?

1, 211

There is a total of 2 positive divisors.
The sum of these divisors is 212.
The arithmetic mean is 106.

2 odd divisors

1, 211

How to compute the divisors of 211?

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 211 by each of the numbers from 1 to 211 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)

211 / 1 = 211 (the remainder is 0, so 1 is a divisor of 211)
211 / 2 = 105.5 (the remainder is 1, so 2 is not a divisor of 211)
211 / 3 = 70.333333333333 (the remainder is 1, so 3 is not a divisor of 211)
...
211 / 210 = 1.0047619047619 (the remainder is 1, so 210 is not a divisor of 211)
211 / 211 = 1 (the remainder is 0, so 211 is a divisor of 211)

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 211 (i.e. 14.525839046334). 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 $\frac{N}{d} = D$ )

211 / 1 = 211 (the remainder is 0, so 1 and 211 are divisors of 211)
211 / 2 = 105.5 (the remainder is 1, so 2 is not a divisor of 211)
211 / 3 = 70.333333333333 (the remainder is 1, so 3 is not a divisor of 211)
...
211 / 13 = 16.230769230769 (the remainder is 3, so 13 is not a divisor of 211)
211 / 14 = 15.071428571429 (the remainder is 1, so 14 is not a divisor of 211)