What are the divisors of 5623?

1, 5623

There is a total of 2 positive divisors.
The sum of these divisors is 5624.
The arithmetic mean is 2812.

2 odd divisors

1, 5623

How to compute the divisors of 5623?

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 5623 by each of the numbers from 1 to 5623 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)

5623 / 1 = 5623 (the remainder is 0, so 1 is a divisor of 5623)
5623 / 2 = 2811.5 (the remainder is 1, so 2 is not a divisor of 5623)
5623 / 3 = 1874.3333333333 (the remainder is 1, so 3 is not a divisor of 5623)
...
5623 / 5622 = 1.0001778726432 (the remainder is 1, so 5622 is not a divisor of 5623)
5623 / 5623 = 1 (the remainder is 0, so 5623 is a divisor of 5623)

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 5623 (i.e. 74.986665481271). 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 $\frac{N}{d} = D$ )

5623 / 1 = 5623 (the remainder is 0, so 1 and 5623 are divisors of 5623)
5623 / 2 = 2811.5 (the remainder is 1, so 2 is not a divisor of 5623)
5623 / 3 = 1874.3333333333 (the remainder is 1, so 3 is not a divisor of 5623)
...
5623 / 73 = 77.027397260274 (the remainder is 2, so 73 is not a divisor of 5623)
5623 / 74 = 75.986486486486 (the remainder is 73, so 74 is not a divisor of 5623)