What are the divisors of 213?

1, 3, 71, 213

There is a total of 4 positive divisors.
The sum of these divisors is 288.
The arithmetic mean is 72.

4 odd divisors

1, 3, 71, 213

How to compute the divisors of 213?

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 213 by each of the numbers from 1 to 213 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)

213 / 1 = 213 (the remainder is 0, so 1 is a divisor of 213)
213 / 2 = 106.5 (the remainder is 1, so 2 is not a divisor of 213)
213 / 3 = 71 (the remainder is 0, so 3 is a divisor of 213)
...
213 / 212 = 1.0047169811321 (the remainder is 1, so 212 is not a divisor of 213)
213 / 213 = 1 (the remainder is 0, so 213 is a divisor of 213)

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 213 (i.e. 14.594519519326). 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 $\frac{N}{d} = D$ )

213 / 1 = 213 (the remainder is 0, so 1 and 213 are divisors of 213)
213 / 2 = 106.5 (the remainder is 1, so 2 is not a divisor of 213)
213 / 3 = 71 (the remainder is 0, so 3 and 71 are divisors of 213)
...
213 / 13 = 16.384615384615 (the remainder is 5, so 13 is not a divisor of 213)
213 / 14 = 15.214285714286 (the remainder is 3, so 14 is not a divisor of 213)