What are the divisors of 5421?

1, 3, 13, 39, 139, 417, 1807, 5421

There is a total of 8 positive divisors.
The sum of these divisors is 7840.
The arithmetic mean is 980.

8 odd divisors

1, 3, 13, 39, 139, 417, 1807, 5421

How to compute the divisors of 5421?

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 5421 by each of the numbers from 1 to 5421 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)

5421 / 1 = 5421 (the remainder is 0, so 1 is a divisor of 5421)
5421 / 2 = 2710.5 (the remainder is 1, so 2 is not a divisor of 5421)
5421 / 3 = 1807 (the remainder is 0, so 3 is a divisor of 5421)
...
5421 / 5420 = 1.000184501845 (the remainder is 1, so 5420 is not a divisor of 5421)
5421 / 5421 = 1 (the remainder is 0, so 5421 is a divisor of 5421)

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 5421 (i.e. 73.6274405368). 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 $\frac{N}{d} = D$ )

5421 / 1 = 5421 (the remainder is 0, so 1 and 5421 are divisors of 5421)
5421 / 2 = 2710.5 (the remainder is 1, so 2 is not a divisor of 5421)
5421 / 3 = 1807 (the remainder is 0, so 3 and 1807 are divisors of 5421)
...
5421 / 72 = 75.291666666667 (the remainder is 21, so 72 is not a divisor of 5421)
5421 / 73 = 74.260273972603 (the remainder is 19, so 73 is not a divisor of 5421)