What are the divisors of 3227?

1, 7, 461, 3227

There is a total of 4 positive divisors.
The sum of these divisors is 3696.
The arithmetic mean is 924.

4 odd divisors

1, 7, 461, 3227

How to compute the divisors of 3227?

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 3227 by each of the numbers from 1 to 3227 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)

3227 / 1 = 3227 (the remainder is 0, so 1 is a divisor of 3227)
3227 / 2 = 1613.5 (the remainder is 1, so 2 is not a divisor of 3227)
3227 / 3 = 1075.6666666667 (the remainder is 2, so 3 is not a divisor of 3227)
...
3227 / 3226 = 1.0003099814011 (the remainder is 1, so 3226 is not a divisor of 3227)
3227 / 3227 = 1 (the remainder is 0, so 3227 is a divisor of 3227)

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 3227 (i.e. 56.806689746895). 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 $\frac{N}{d} = D$ )

3227 / 1 = 3227 (the remainder is 0, so 1 and 3227 are divisors of 3227)
3227 / 2 = 1613.5 (the remainder is 1, so 2 is not a divisor of 3227)
3227 / 3 = 1075.6666666667 (the remainder is 2, so 3 is not a divisor of 3227)
...
3227 / 55 = 58.672727272727 (the remainder is 37, so 55 is not a divisor of 3227)
3227 / 56 = 57.625 (the remainder is 35, so 56 is not a divisor of 3227)