What are the divisors of 1310?

1, 2, 5, 10, 131, 262, 655, 1310

There is a total of 8 positive divisors.
The sum of these divisors is 2376.
The arithmetic mean is 297.

4 even divisors

2, 10, 262, 1310

4 odd divisors

1, 5, 131, 655

How to compute the divisors of 1310?

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 1310 by each of the numbers from 1 to 1310 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)

1310 / 1 = 1310 (the remainder is 0, so 1 is a divisor of 1310)
1310 / 2 = 655 (the remainder is 0, so 2 is a divisor of 1310)
1310 / 3 = 436.66666666667 (the remainder is 2, so 3 is not a divisor of 1310)
...
1310 / 1309 = 1.0007639419404 (the remainder is 1, so 1309 is not a divisor of 1310)
1310 / 1310 = 1 (the remainder is 0, so 1310 is a divisor of 1310)

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 1310 (i.e. 36.193922141708). 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 $\frac{N}{d} = D$ )

1310 / 1 = 1310 (the remainder is 0, so 1 and 1310 are divisors of 1310)
1310 / 2 = 655 (the remainder is 0, so 2 and 655 are divisors of 1310)
1310 / 3 = 436.66666666667 (the remainder is 2, so 3 is not a divisor of 1310)
...
1310 / 35 = 37.428571428571 (the remainder is 15, so 35 is not a divisor of 1310)
1310 / 36 = 36.388888888889 (the remainder is 14, so 36 is not a divisor of 1310)