What are the divisors of 1427?

1, 1427

There is a total of 2 positive divisors.
The sum of these divisors is 1428.
The arithmetic mean is 714.

2 odd divisors

1, 1427

How to compute the divisors of 1427?

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 1427 by each of the numbers from 1 to 1427 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)

1427 / 1 = 1427 (the remainder is 0, so 1 is a divisor of 1427)
1427 / 2 = 713.5 (the remainder is 1, so 2 is not a divisor of 1427)
1427 / 3 = 475.66666666667 (the remainder is 2, so 3 is not a divisor of 1427)
...
1427 / 1426 = 1.0007012622721 (the remainder is 1, so 1426 is not a divisor of 1427)
1427 / 1427 = 1 (the remainder is 0, so 1427 is a divisor of 1427)

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 1427 (i.e. 37.775653535048). 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 $\frac{N}{d} = D$ )

1427 / 1 = 1427 (the remainder is 0, so 1 and 1427 are divisors of 1427)
1427 / 2 = 713.5 (the remainder is 1, so 2 is not a divisor of 1427)
1427 / 3 = 475.66666666667 (the remainder is 2, so 3 is not a divisor of 1427)
...
1427 / 36 = 39.638888888889 (the remainder is 23, so 36 is not a divisor of 1427)
1427 / 37 = 38.567567567568 (the remainder is 21, so 37 is not a divisor of 1427)