What are the divisors of 201?

1, 3, 67, 201

There is a total of 4 positive divisors.
The sum of these divisors is 272.
The arithmetic mean is 68.

4 odd divisors

1, 3, 67, 201

How to compute the divisors of 201?

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 201 by each of the numbers from 1 to 201 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)

201 / 1 = 201 (the remainder is 0, so 1 is a divisor of 201)
201 / 2 = 100.5 (the remainder is 1, so 2 is not a divisor of 201)
201 / 3 = 67 (the remainder is 0, so 3 is a divisor of 201)
...
201 / 200 = 1.005 (the remainder is 1, so 200 is not a divisor of 201)
201 / 201 = 1 (the remainder is 0, so 201 is a divisor of 201)

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 201 (i.e. 14.177446878758). 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 $\frac{N}{d} = D$ )

201 / 1 = 201 (the remainder is 0, so 1 and 201 are divisors of 201)
201 / 2 = 100.5 (the remainder is 1, so 2 is not a divisor of 201)
201 / 3 = 67 (the remainder is 0, so 3 and 67 are divisors of 201)
...
201 / 13 = 15.461538461538 (the remainder is 6, so 13 is not a divisor of 201)
201 / 14 = 14.357142857143 (the remainder is 5, so 14 is not a divisor of 201)