What are the divisors of 1563?

1, 3, 521, 1563

There is a total of 4 positive divisors.
The sum of these divisors is 2088.
The arithmetic mean is 522.

4 odd divisors

1, 3, 521, 1563

How to compute the divisors of 1563?

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 1563 by each of the numbers from 1 to 1563 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)

1563 / 1 = 1563 (the remainder is 0, so 1 is a divisor of 1563)
1563 / 2 = 781.5 (the remainder is 1, so 2 is not a divisor of 1563)
1563 / 3 = 521 (the remainder is 0, so 3 is a divisor of 1563)
...
1563 / 1562 = 1.0006402048656 (the remainder is 1, so 1562 is not a divisor of 1563)
1563 / 1563 = 1 (the remainder is 0, so 1563 is a divisor of 1563)

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 1563 (i.e. 39.534794801542). 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 $\frac{N}{d} = D$ )

1563 / 1 = 1563 (the remainder is 0, so 1 and 1563 are divisors of 1563)
1563 / 2 = 781.5 (the remainder is 1, so 2 is not a divisor of 1563)
1563 / 3 = 521 (the remainder is 0, so 3 and 521 are divisors of 1563)
...
1563 / 38 = 41.131578947368 (the remainder is 5, so 38 is not a divisor of 1563)
1563 / 39 = 40.076923076923 (the remainder is 3, so 39 is not a divisor of 1563)