What are the divisors of 1399?

1, 1399

There is a total of 2 positive divisors.
The sum of these divisors is 1400.
The arithmetic mean is 700.

2 odd divisors

1, 1399

How to compute the divisors of 1399?

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 1399 by each of the numbers from 1 to 1399 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)

1399 / 1 = 1399 (the remainder is 0, so 1 is a divisor of 1399)
1399 / 2 = 699.5 (the remainder is 1, so 2 is not a divisor of 1399)
1399 / 3 = 466.33333333333 (the remainder is 1, so 3 is not a divisor of 1399)
...
1399 / 1398 = 1.0007153075823 (the remainder is 1, so 1398 is not a divisor of 1399)
1399 / 1399 = 1 (the remainder is 0, so 1399 is a divisor of 1399)

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 1399 (i.e. 37.40320841853). 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 $\frac{N}{d} = D$ )

1399 / 1 = 1399 (the remainder is 0, so 1 and 1399 are divisors of 1399)
1399 / 2 = 699.5 (the remainder is 1, so 2 is not a divisor of 1399)
1399 / 3 = 466.33333333333 (the remainder is 1, so 3 is not a divisor of 1399)
...
1399 / 36 = 38.861111111111 (the remainder is 31, so 36 is not a divisor of 1399)
1399 / 37 = 37.810810810811 (the remainder is 30, so 37 is not a divisor of 1399)