What are the divisors of 1315?

1, 5, 263, 1315

There is a total of 4 positive divisors.
The sum of these divisors is 1584.
The arithmetic mean is 396.

4 odd divisors

1, 5, 263, 1315

How to compute the divisors of 1315?

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 1315 by each of the numbers from 1 to 1315 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)

1315 / 1 = 1315 (the remainder is 0, so 1 is a divisor of 1315)
1315 / 2 = 657.5 (the remainder is 1, so 2 is not a divisor of 1315)
1315 / 3 = 438.33333333333 (the remainder is 1, so 3 is not a divisor of 1315)
...
1315 / 1314 = 1.0007610350076 (the remainder is 1, so 1314 is not a divisor of 1315)
1315 / 1315 = 1 (the remainder is 0, so 1315 is a divisor of 1315)

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 1315 (i.e. 36.262928728937). 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 $\frac{N}{d} = D$ )

1315 / 1 = 1315 (the remainder is 0, so 1 and 1315 are divisors of 1315)
1315 / 2 = 657.5 (the remainder is 1, so 2 is not a divisor of 1315)
1315 / 3 = 438.33333333333 (the remainder is 1, so 3 is not a divisor of 1315)
...
1315 / 35 = 37.571428571429 (the remainder is 20, so 35 is not a divisor of 1315)
1315 / 36 = 36.527777777778 (the remainder is 19, so 36 is not a divisor of 1315)