What are the divisors of 5460?

1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 20, 21, 26, 28, 30, 35, 39, 42, 52, 60, 65, 70, 78, 84, 91, 105, 130, 140, 156, 182, 195, 210, 260, 273, 364, 390, 420, 455, 546, 780, 910, 1092, 1365, 1820, 2730, 5460

There is a total of 48 positive divisors.
The sum of these divisors is 18816.
The arithmetic mean is 392.

32 even divisors

2, 4, 6, 10, 12, 14, 20, 26, 28, 30, 42, 52, 60, 70, 78, 84, 130, 140, 156, 182, 210, 260, 364, 390, 420, 546, 780, 910, 1092, 1820, 2730, 5460

16 odd divisors

1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365

How to compute the divisors of 5460?

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 5460 by each of the numbers from 1 to 5460 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)

5460 / 1 = 5460 (the remainder is 0, so 1 is a divisor of 5460)
5460 / 2 = 2730 (the remainder is 0, so 2 is a divisor of 5460)
5460 / 3 = 1820 (the remainder is 0, so 3 is a divisor of 5460)
...
5460 / 5459 = 1.0001831837333 (the remainder is 1, so 5459 is not a divisor of 5460)
5460 / 5460 = 1 (the remainder is 0, so 5460 is a divisor of 5460)

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 5460 (i.e. 73.891812807645). 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 $\frac{N}{d} = D$ )

5460 / 1 = 5460 (the remainder is 0, so 1 and 5460 are divisors of 5460)
5460 / 2 = 2730 (the remainder is 0, so 2 and 2730 are divisors of 5460)
5460 / 3 = 1820 (the remainder is 0, so 3 and 1820 are divisors of 5460)
...
5460 / 72 = 75.833333333333 (the remainder is 60, so 72 is not a divisor of 5460)
5460 / 73 = 74.794520547945 (the remainder is 58, so 73 is not a divisor of 5460)