What are the divisors of 1369?

1, 37, 1369

There is a total of 3 positive divisors.
The sum of these divisors is 1407.
The arithmetic mean is 469.

3 odd divisors

1, 37, 1369

How to compute the divisors of 1369?

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 1369 by each of the numbers from 1 to 1369 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)

1369 / 1 = 1369 (the remainder is 0, so 1 is a divisor of 1369)
1369 / 2 = 684.5 (the remainder is 1, so 2 is not a divisor of 1369)
1369 / 3 = 456.33333333333 (the remainder is 1, so 3 is not a divisor of 1369)
...
1369 / 1368 = 1.000730994152 (the remainder is 1, so 1368 is not a divisor of 1369)
1369 / 1369 = 1 (the remainder is 0, so 1369 is a divisor of 1369)

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 1369 (i.e. 37). 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 $\frac{N}{d} = D$ )

1369 / 1 = 1369 (the remainder is 0, so 1 and 1369 are divisors of 1369)
1369 / 2 = 684.5 (the remainder is 1, so 2 is not a divisor of 1369)
1369 / 3 = 456.33333333333 (the remainder is 1, so 3 is not a divisor of 1369)
...
1369 / 36 = 38.027777777778 (the remainder is 1, so 36 is not a divisor of 1369)
1369 / 37 = 37 (the remainder is 0, so 37 and 37 are divisors of 1369)