What are the divisors of 3221?

1, 3221

There is a total of 2 positive divisors.
The sum of these divisors is 3222.
The arithmetic mean is 1611.

2 odd divisors

1, 3221

How to compute the divisors of 3221?

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 3221 by each of the numbers from 1 to 3221 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)

3221 / 1 = 3221 (the remainder is 0, so 1 is a divisor of 3221)
3221 / 2 = 1610.5 (the remainder is 1, so 2 is not a divisor of 3221)
3221 / 3 = 1073.6666666667 (the remainder is 2, so 3 is not a divisor of 3221)
...
3221 / 3220 = 1.0003105590062 (the remainder is 1, so 3220 is not a divisor of 3221)
3221 / 3221 = 1 (the remainder is 0, so 3221 is a divisor of 3221)

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 3221 (i.e. 56.753854494651). 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 $\frac{N}{d} = D$ )

3221 / 1 = 3221 (the remainder is 0, so 1 and 3221 are divisors of 3221)
3221 / 2 = 1610.5 (the remainder is 1, so 2 is not a divisor of 3221)
3221 / 3 = 1073.6666666667 (the remainder is 2, so 3 is not a divisor of 3221)
...
3221 / 55 = 58.563636363636 (the remainder is 31, so 55 is not a divisor of 3221)
3221 / 56 = 57.517857142857 (the remainder is 29, so 56 is not a divisor of 3221)