What are the divisors of 1567?

1, 1567

There is a total of 2 positive divisors.
The sum of these divisors is 1568.
The arithmetic mean is 784.

2 odd divisors

1, 1567

How to compute the divisors of 1567?

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 1567 by each of the numbers from 1 to 1567 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)

1567 / 1 = 1567 (the remainder is 0, so 1 is a divisor of 1567)
1567 / 2 = 783.5 (the remainder is 1, so 2 is not a divisor of 1567)
1567 / 3 = 522.33333333333 (the remainder is 1, so 3 is not a divisor of 1567)
...
1567 / 1566 = 1.0006385696041 (the remainder is 1, so 1566 is not a divisor of 1567)
1567 / 1567 = 1 (the remainder is 0, so 1567 is a divisor of 1567)

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 1567 (i.e. 39.585350825779). 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 $\frac{N}{d} = D$ )

1567 / 1 = 1567 (the remainder is 0, so 1 and 1567 are divisors of 1567)
1567 / 2 = 783.5 (the remainder is 1, so 2 is not a divisor of 1567)
1567 / 3 = 522.33333333333 (the remainder is 1, so 3 is not a divisor of 1567)
...
1567 / 38 = 41.236842105263 (the remainder is 9, so 38 is not a divisor of 1567)
1567 / 39 = 40.179487179487 (the remainder is 7, so 39 is not a divisor of 1567)