What are the divisors of 9221?

1, 9221

There is a total of 2 positive divisors.
The sum of these divisors is 9222.
The arithmetic mean is 4611.

2 odd divisors

1, 9221

How to compute the divisors of 9221?

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 9221 by each of the numbers from 1 to 9221 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)

9221 / 1 = 9221 (the remainder is 0, so 1 is a divisor of 9221)
9221 / 2 = 4610.5 (the remainder is 1, so 2 is not a divisor of 9221)
9221 / 3 = 3073.6666666667 (the remainder is 2, so 3 is not a divisor of 9221)
...
9221 / 9220 = 1.0001084598698 (the remainder is 1, so 9220 is not a divisor of 9221)
9221 / 9221 = 1 (the remainder is 0, so 9221 is a divisor of 9221)

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 9221 (i.e. 96.026038135497). 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 $\frac{N}{d} = D$ )

9221 / 1 = 9221 (the remainder is 0, so 1 and 9221 are divisors of 9221)
9221 / 2 = 4610.5 (the remainder is 1, so 2 is not a divisor of 9221)
9221 / 3 = 3073.6666666667 (the remainder is 2, so 3 is not a divisor of 9221)
...
9221 / 95 = 97.063157894737 (the remainder is 6, so 95 is not a divisor of 9221)
9221 / 96 = 96.052083333333 (the remainder is 5, so 96 is not a divisor of 9221)