What are the divisors of 1367?

1, 1367

There is a total of 2 positive divisors.
The sum of these divisors is 1368.
The arithmetic mean is 684.

2 odd divisors

1, 1367

How to compute the divisors of 1367?

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 1367 by each of the numbers from 1 to 1367 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)

1367 / 1 = 1367 (the remainder is 0, so 1 is a divisor of 1367)
1367 / 2 = 683.5 (the remainder is 1, so 2 is not a divisor of 1367)
1367 / 3 = 455.66666666667 (the remainder is 2, so 3 is not a divisor of 1367)
...
1367 / 1366 = 1.0007320644217 (the remainder is 1, so 1366 is not a divisor of 1367)
1367 / 1367 = 1 (the remainder is 0, so 1367 is a divisor of 1367)

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 1367 (i.e. 36.972963094672). 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 $\frac{N}{d} = D$ )

1367 / 1 = 1367 (the remainder is 0, so 1 and 1367 are divisors of 1367)
1367 / 2 = 683.5 (the remainder is 1, so 2 is not a divisor of 1367)
1367 / 3 = 455.66666666667 (the remainder is 2, so 3 is not a divisor of 1367)
...
1367 / 35 = 39.057142857143 (the remainder is 2, so 35 is not a divisor of 1367)
1367 / 36 = 37.972222222222 (the remainder is 35, so 36 is not a divisor of 1367)