What are the divisors of 5197?

1, 5197

There is a total of 2 positive divisors.
The sum of these divisors is 5198.
The arithmetic mean is 2599.

2 odd divisors

1, 5197

How to compute the divisors of 5197?

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 5197 by each of the numbers from 1 to 5197 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)

5197 / 1 = 5197 (the remainder is 0, so 1 is a divisor of 5197)
5197 / 2 = 2598.5 (the remainder is 1, so 2 is not a divisor of 5197)
5197 / 3 = 1732.3333333333 (the remainder is 1, so 3 is not a divisor of 5197)
...
5197 / 5196 = 1.0001924557352 (the remainder is 1, so 5196 is not a divisor of 5197)
5197 / 5197 = 1 (the remainder is 0, so 5197 is a divisor of 5197)

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 5197 (i.e. 72.090221250874). 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 $\frac{N}{d} = D$ )

5197 / 1 = 5197 (the remainder is 0, so 1 and 5197 are divisors of 5197)
5197 / 2 = 2598.5 (the remainder is 1, so 2 is not a divisor of 5197)
5197 / 3 = 1732.3333333333 (the remainder is 1, so 3 is not a divisor of 5197)
...
5197 / 71 = 73.197183098592 (the remainder is 14, so 71 is not a divisor of 5197)
5197 / 72 = 72.180555555556 (the remainder is 13, so 72 is not a divisor of 5197)