What are the divisors of 1307?

1, 1307

There is a total of 2 positive divisors.
The sum of these divisors is 1308.
The arithmetic mean is 654.

2 odd divisors

1, 1307

How to compute the divisors of 1307?

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 1307 by each of the numbers from 1 to 1307 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)

1307 / 1 = 1307 (the remainder is 0, so 1 is a divisor of 1307)
1307 / 2 = 653.5 (the remainder is 1, so 2 is not a divisor of 1307)
1307 / 3 = 435.66666666667 (the remainder is 2, so 3 is not a divisor of 1307)
...
1307 / 1306 = 1.0007656967841 (the remainder is 1, so 1306 is not a divisor of 1307)
1307 / 1307 = 1 (the remainder is 0, so 1307 is a divisor of 1307)

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 1307 (i.e. 36.15245496505). 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 $\frac{N}{d} = D$ )

1307 / 1 = 1307 (the remainder is 0, so 1 and 1307 are divisors of 1307)
1307 / 2 = 653.5 (the remainder is 1, so 2 is not a divisor of 1307)
1307 / 3 = 435.66666666667 (the remainder is 2, so 3 is not a divisor of 1307)
...
1307 / 35 = 37.342857142857 (the remainder is 12, so 35 is not a divisor of 1307)
1307 / 36 = 36.305555555556 (the remainder is 11, so 36 is not a divisor of 1307)