What are the divisors of 1423?

1, 1423

There is a total of 2 positive divisors.
The sum of these divisors is 1424.
The arithmetic mean is 712.

2 odd divisors

1, 1423

How to compute the divisors of 1423?

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 1423 by each of the numbers from 1 to 1423 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)

1423 / 1 = 1423 (the remainder is 0, so 1 is a divisor of 1423)
1423 / 2 = 711.5 (the remainder is 1, so 2 is not a divisor of 1423)
1423 / 3 = 474.33333333333 (the remainder is 1, so 3 is not a divisor of 1423)
...
1423 / 1422 = 1.0007032348805 (the remainder is 1, so 1422 is not a divisor of 1423)
1423 / 1423 = 1 (the remainder is 0, so 1423 is a divisor of 1423)

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 1423 (i.e. 37.72267222772). 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 $\frac{N}{d} = D$ )

1423 / 1 = 1423 (the remainder is 0, so 1 and 1423 are divisors of 1423)
1423 / 2 = 711.5 (the remainder is 1, so 2 is not a divisor of 1423)
1423 / 3 = 474.33333333333 (the remainder is 1, so 3 is not a divisor of 1423)
...
1423 / 36 = 39.527777777778 (the remainder is 19, so 36 is not a divisor of 1423)
1423 / 37 = 38.459459459459 (the remainder is 17, so 37 is not a divisor of 1423)