What are the divisors of 1428?

1, 2, 3, 4, 6, 7, 12, 14, 17, 21, 28, 34, 42, 51, 68, 84, 102, 119, 204, 238, 357, 476, 714, 1428

There is a total of 24 positive divisors.
The sum of these divisors is 4032.
The arithmetic mean is 168.

16 even divisors

2, 4, 6, 12, 14, 28, 34, 42, 68, 84, 102, 204, 238, 476, 714, 1428

8 odd divisors

1, 3, 7, 17, 21, 51, 119, 357

How to compute the divisors of 1428?

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 1428 by each of the numbers from 1 to 1428 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)

1428 / 1 = 1428 (the remainder is 0, so 1 is a divisor of 1428)
1428 / 2 = 714 (the remainder is 0, so 2 is a divisor of 1428)
1428 / 3 = 476 (the remainder is 0, so 3 is a divisor of 1428)
...
1428 / 1427 = 1.0007007708479 (the remainder is 1, so 1427 is not a divisor of 1428)
1428 / 1428 = 1 (the remainder is 0, so 1428 is a divisor of 1428)

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 1428 (i.e. 37.788887255382). 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 $\frac{N}{d} = D$ )

1428 / 1 = 1428 (the remainder is 0, so 1 and 1428 are divisors of 1428)
1428 / 2 = 714 (the remainder is 0, so 2 and 714 are divisors of 1428)
1428 / 3 = 476 (the remainder is 0, so 3 and 476 are divisors of 1428)
...
1428 / 36 = 39.666666666667 (the remainder is 24, so 36 is not a divisor of 1428)
1428 / 37 = 38.594594594595 (the remainder is 22, so 37 is not a divisor of 1428)