What are the divisors of 3491?

1, 3491

There is a total of 2 positive divisors.
The sum of these divisors is 3492.
The arithmetic mean is 1746.

2 odd divisors

1, 3491

How to compute the divisors of 3491?

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 3491 by each of the numbers from 1 to 3491 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)

3491 / 1 = 3491 (the remainder is 0, so 1 is a divisor of 3491)
3491 / 2 = 1745.5 (the remainder is 1, so 2 is not a divisor of 3491)
3491 / 3 = 1163.6666666667 (the remainder is 2, so 3 is not a divisor of 3491)
...
3491 / 3490 = 1.0002865329513 (the remainder is 1, so 3490 is not a divisor of 3491)
3491 / 3491 = 1 (the remainder is 0, so 3491 is a divisor of 3491)

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 3491 (i.e. 59.08468498689). 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 $\frac{N}{d} = D$ )

3491 / 1 = 3491 (the remainder is 0, so 1 and 3491 are divisors of 3491)
3491 / 2 = 1745.5 (the remainder is 1, so 2 is not a divisor of 3491)
3491 / 3 = 1163.6666666667 (the remainder is 2, so 3 is not a divisor of 3491)
...
3491 / 58 = 60.189655172414 (the remainder is 11, so 58 is not a divisor of 3491)
3491 / 59 = 59.169491525424 (the remainder is 10, so 59 is not a divisor of 3491)