What are the divisors of 1306?

1, 2, 653, 1306

There is a total of 4 positive divisors.
The sum of these divisors is 1962.
The arithmetic mean is 490.5.

2 even divisors

2, 1306

2 odd divisors

1, 653

How to compute the divisors of 1306?

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 1306 by each of the numbers from 1 to 1306 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)

1306 / 1 = 1306 (the remainder is 0, so 1 is a divisor of 1306)
1306 / 2 = 653 (the remainder is 0, so 2 is a divisor of 1306)
1306 / 3 = 435.33333333333 (the remainder is 1, so 3 is not a divisor of 1306)
...
1306 / 1305 = 1.0007662835249 (the remainder is 1, so 1305 is not a divisor of 1306)
1306 / 1306 = 1 (the remainder is 0, so 1306 is a divisor of 1306)

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 1306 (i.e. 36.138621999185). 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 $\frac{N}{d} = D$ )

1306 / 1 = 1306 (the remainder is 0, so 1 and 1306 are divisors of 1306)
1306 / 2 = 653 (the remainder is 0, so 2 and 653 are divisors of 1306)
1306 / 3 = 435.33333333333 (the remainder is 1, so 3 is not a divisor of 1306)
...
1306 / 35 = 37.314285714286 (the remainder is 11, so 35 is not a divisor of 1306)
1306 / 36 = 36.277777777778 (the remainder is 10, so 36 is not a divisor of 1306)