What are the divisors of 3251?

1, 3251

There is a total of 2 positive divisors.
The sum of these divisors is 3252.
The arithmetic mean is 1626.

2 odd divisors

1, 3251

How to compute the divisors of 3251?

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 3251 by each of the numbers from 1 to 3251 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)

3251 / 1 = 3251 (the remainder is 0, so 1 is a divisor of 3251)
3251 / 2 = 1625.5 (the remainder is 1, so 2 is not a divisor of 3251)
3251 / 3 = 1083.6666666667 (the remainder is 2, so 3 is not a divisor of 3251)
...
3251 / 3250 = 1.0003076923077 (the remainder is 1, so 3250 is not a divisor of 3251)
3251 / 3251 = 1 (the remainder is 0, so 3251 is a divisor of 3251)

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 3251 (i.e. 57.017541160594). 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 $\frac{N}{d} = D$ )

3251 / 1 = 3251 (the remainder is 0, so 1 and 3251 are divisors of 3251)
3251 / 2 = 1625.5 (the remainder is 1, so 2 is not a divisor of 3251)
3251 / 3 = 1083.6666666667 (the remainder is 2, so 3 is not a divisor of 3251)
...
3251 / 56 = 58.053571428571 (the remainder is 3, so 56 is not a divisor of 3251)
3251 / 57 = 57.035087719298 (the remainder is 2, so 57 is not a divisor of 3251)