What are the divisors of 9191?

1, 7, 13, 91, 101, 707, 1313, 9191

There is a total of 8 positive divisors.
The sum of these divisors is 11424.
The arithmetic mean is 1428.

8 odd divisors

1, 7, 13, 91, 101, 707, 1313, 9191

How to compute the divisors of 9191?

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 9191 by each of the numbers from 1 to 9191 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)

9191 / 1 = 9191 (the remainder is 0, so 1 is a divisor of 9191)
9191 / 2 = 4595.5 (the remainder is 1, so 2 is not a divisor of 9191)
9191 / 3 = 3063.6666666667 (the remainder is 2, so 3 is not a divisor of 9191)
...
9191 / 9190 = 1.0001088139282 (the remainder is 1, so 9190 is not a divisor of 9191)
9191 / 9191 = 1 (the remainder is 0, so 9191 is a divisor of 9191)

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 9191 (i.e. 95.869703243517). 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 $\frac{N}{d} = D$ )

9191 / 1 = 9191 (the remainder is 0, so 1 and 9191 are divisors of 9191)
9191 / 2 = 4595.5 (the remainder is 1, so 2 is not a divisor of 9191)
9191 / 3 = 3063.6666666667 (the remainder is 2, so 3 is not a divisor of 9191)
...
9191 / 94 = 97.776595744681 (the remainder is 73, so 94 is not a divisor of 9191)
9191 / 95 = 96.747368421053 (the remainder is 71, so 95 is not a divisor of 9191)