What are the divisors of 3211?

1, 13, 19, 169, 247, 3211

There is a total of 6 positive divisors.
The sum of these divisors is 3660.
The arithmetic mean is 610.

6 odd divisors

1, 13, 19, 169, 247, 3211

How to compute the divisors of 3211?

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 3211 by each of the numbers from 1 to 3211 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)

3211 / 1 = 3211 (the remainder is 0, so 1 is a divisor of 3211)
3211 / 2 = 1605.5 (the remainder is 1, so 2 is not a divisor of 3211)
3211 / 3 = 1070.3333333333 (the remainder is 1, so 3 is not a divisor of 3211)
...
3211 / 3210 = 1.0003115264798 (the remainder is 1, so 3210 is not a divisor of 3211)
3211 / 3211 = 1 (the remainder is 0, so 3211 is a divisor of 3211)

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 3211 (i.e. 56.665686266029). 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 $\frac{N}{d} = D$ )

3211 / 1 = 3211 (the remainder is 0, so 1 and 3211 are divisors of 3211)
3211 / 2 = 1605.5 (the remainder is 1, so 2 is not a divisor of 3211)
3211 / 3 = 1070.3333333333 (the remainder is 1, so 3 is not a divisor of 3211)
...
3211 / 55 = 58.381818181818 (the remainder is 21, so 55 is not a divisor of 3211)
3211 / 56 = 57.339285714286 (the remainder is 19, so 56 is not a divisor of 3211)