What are the divisors of 221?

1, 13, 17, 221

There is a total of 4 positive divisors.
The sum of these divisors is 252.
The arithmetic mean is 63.

4 odd divisors

1, 13, 17, 221

How to compute the divisors of 221?

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 221 by each of the numbers from 1 to 221 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)

221 / 1 = 221 (the remainder is 0, so 1 is a divisor of 221)
221 / 2 = 110.5 (the remainder is 1, so 2 is not a divisor of 221)
221 / 3 = 73.666666666667 (the remainder is 2, so 3 is not a divisor of 221)
...
221 / 220 = 1.0045454545455 (the remainder is 1, so 220 is not a divisor of 221)
221 / 221 = 1 (the remainder is 0, so 221 is a divisor of 221)

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 221 (i.e. 14.866068747319). 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 $\frac{N}{d} = D$ )

221 / 1 = 221 (the remainder is 0, so 1 and 221 are divisors of 221)
221 / 2 = 110.5 (the remainder is 1, so 2 is not a divisor of 221)
221 / 3 = 73.666666666667 (the remainder is 2, so 3 is not a divisor of 221)
...
221 / 13 = 17 (the remainder is 0, so 13 and 17 are divisors of 221)
221 / 14 = 15.785714285714 (the remainder is 11, so 14 is not a divisor of 221)