What are the divisors of 3461?

1, 3461

There is a total of 2 positive divisors.
The sum of these divisors is 3462.
The arithmetic mean is 1731.

2 odd divisors

1, 3461

How to compute the divisors of 3461?

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 3461 by each of the numbers from 1 to 3461 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)

3461 / 1 = 3461 (the remainder is 0, so 1 is a divisor of 3461)
3461 / 2 = 1730.5 (the remainder is 1, so 2 is not a divisor of 3461)
3461 / 3 = 1153.6666666667 (the remainder is 2, so 3 is not a divisor of 3461)
...
3461 / 3460 = 1.000289017341 (the remainder is 1, so 3460 is not a divisor of 3461)
3461 / 3461 = 1 (the remainder is 0, so 3461 is a divisor of 3461)

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 3461 (i.e. 58.830264320331). 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 $\frac{N}{d} = D$ )

3461 / 1 = 3461 (the remainder is 0, so 1 and 3461 are divisors of 3461)
3461 / 2 = 1730.5 (the remainder is 1, so 2 is not a divisor of 3461)
3461 / 3 = 1153.6666666667 (the remainder is 2, so 3 is not a divisor of 3461)
...
3461 / 57 = 60.719298245614 (the remainder is 41, so 57 is not a divisor of 3461)
3461 / 58 = 59.672413793103 (the remainder is 39, so 58 is not a divisor of 3461)