What are the divisors of 3421?

1, 11, 311, 3421

There is a total of 4 positive divisors.
The sum of these divisors is 3744.
The arithmetic mean is 936.

4 odd divisors

1, 11, 311, 3421

How to compute the divisors of 3421?

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 3421 by each of the numbers from 1 to 3421 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)

3421 / 1 = 3421 (the remainder is 0, so 1 is a divisor of 3421)
3421 / 2 = 1710.5 (the remainder is 1, so 2 is not a divisor of 3421)
3421 / 3 = 1140.3333333333 (the remainder is 1, so 3 is not a divisor of 3421)
...
3421 / 3420 = 1.0002923976608 (the remainder is 1, so 3420 is not a divisor of 3421)
3421 / 3421 = 1 (the remainder is 0, so 3421 is a divisor of 3421)

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 3421 (i.e. 58.489315263559). 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 $\frac{N}{d} = D$ )

3421 / 1 = 3421 (the remainder is 0, so 1 and 3421 are divisors of 3421)
3421 / 2 = 1710.5 (the remainder is 1, so 2 is not a divisor of 3421)
3421 / 3 = 1140.3333333333 (the remainder is 1, so 3 is not a divisor of 3421)
...
3421 / 57 = 60.017543859649 (the remainder is 1, so 57 is not a divisor of 3421)
3421 / 58 = 58.98275862069 (the remainder is 57, so 58 is not a divisor of 3421)