What are the divisors of 1565?

1, 5, 313, 1565

There is a total of 4 positive divisors.
The sum of these divisors is 1884.
The arithmetic mean is 471.

4 odd divisors

1, 5, 313, 1565

How to compute the divisors of 1565?

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 1565 by each of the numbers from 1 to 1565 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)

1565 / 1 = 1565 (the remainder is 0, so 1 is a divisor of 1565)
1565 / 2 = 782.5 (the remainder is 1, so 2 is not a divisor of 1565)
1565 / 3 = 521.66666666667 (the remainder is 2, so 3 is not a divisor of 1565)
...
1565 / 1564 = 1.0006393861893 (the remainder is 1, so 1564 is not a divisor of 1565)
1565 / 1565 = 1 (the remainder is 0, so 1565 is a divisor of 1565)

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 1565 (i.e. 39.560080889705). 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 $\frac{N}{d} = D$ )

1565 / 1 = 1565 (the remainder is 0, so 1 and 1565 are divisors of 1565)
1565 / 2 = 782.5 (the remainder is 1, so 2 is not a divisor of 1565)
1565 / 3 = 521.66666666667 (the remainder is 2, so 3 is not a divisor of 1565)
...
1565 / 38 = 41.184210526316 (the remainder is 7, so 38 is not a divisor of 1565)
1565 / 39 = 40.128205128205 (the remainder is 5, so 39 is not a divisor of 1565)