What are the divisors of 1397?

1, 11, 127, 1397

There is a total of 4 positive divisors.
The sum of these divisors is 1536.
The arithmetic mean is 384.

4 odd divisors

1, 11, 127, 1397

How to compute the divisors of 1397?

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 1397 by each of the numbers from 1 to 1397 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)

1397 / 1 = 1397 (the remainder is 0, so 1 is a divisor of 1397)
1397 / 2 = 698.5 (the remainder is 1, so 2 is not a divisor of 1397)
1397 / 3 = 465.66666666667 (the remainder is 2, so 3 is not a divisor of 1397)
...
1397 / 1396 = 1.0007163323782 (the remainder is 1, so 1396 is not a divisor of 1397)
1397 / 1397 = 1 (the remainder is 0, so 1397 is a divisor of 1397)

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 1397 (i.e. 37.376463182062). 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 $\frac{N}{d} = D$ )

1397 / 1 = 1397 (the remainder is 0, so 1 and 1397 are divisors of 1397)
1397 / 2 = 698.5 (the remainder is 1, so 2 is not a divisor of 1397)
1397 / 3 = 465.66666666667 (the remainder is 2, so 3 is not a divisor of 1397)
...
1397 / 36 = 38.805555555556 (the remainder is 29, so 36 is not a divisor of 1397)
1397 / 37 = 37.756756756757 (the remainder is 28, so 37 is not a divisor of 1397)