What are the divisors of 5419?

1, 5419

There is a total of 2 positive divisors.
The sum of these divisors is 5420.
The arithmetic mean is 2710.

2 odd divisors

1, 5419

How to compute the divisors of 5419?

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 5419 by each of the numbers from 1 to 5419 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)

5419 / 1 = 5419 (the remainder is 0, so 1 is a divisor of 5419)
5419 / 2 = 2709.5 (the remainder is 1, so 2 is not a divisor of 5419)
5419 / 3 = 1806.3333333333 (the remainder is 1, so 3 is not a divisor of 5419)
...
5419 / 5418 = 1.000184569952 (the remainder is 1, so 5418 is not a divisor of 5419)
5419 / 5419 = 1 (the remainder is 0, so 5419 is a divisor of 5419)

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 5419 (i.e. 73.613857391119). 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 $\frac{N}{d} = D$ )

5419 / 1 = 5419 (the remainder is 0, so 1 and 5419 are divisors of 5419)
5419 / 2 = 2709.5 (the remainder is 1, so 2 is not a divisor of 5419)
5419 / 3 = 1806.3333333333 (the remainder is 1, so 3 is not a divisor of 5419)
...
5419 / 72 = 75.263888888889 (the remainder is 19, so 72 is not a divisor of 5419)
5419 / 73 = 74.232876712329 (the remainder is 17, so 73 is not a divisor of 5419)