What are the divisors of 4561?

1, 4561

There is a total of 2 positive divisors.
The sum of these divisors is 4562.
The arithmetic mean is 2281.

2 odd divisors

1, 4561

How to compute the divisors of 4561?

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 4561 by each of the numbers from 1 to 4561 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)

4561 / 1 = 4561 (the remainder is 0, so 1 is a divisor of 4561)
4561 / 2 = 2280.5 (the remainder is 1, so 2 is not a divisor of 4561)
4561 / 3 = 1520.3333333333 (the remainder is 1, so 3 is not a divisor of 4561)
...
4561 / 4560 = 1.0002192982456 (the remainder is 1, so 4560 is not a divisor of 4561)
4561 / 4561 = 1 (the remainder is 0, so 4561 is a divisor of 4561)

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 4561 (i.e. 67.535176019612). 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 $\frac{N}{d} = D$ )

4561 / 1 = 4561 (the remainder is 0, so 1 and 4561 are divisors of 4561)
4561 / 2 = 2280.5 (the remainder is 1, so 2 is not a divisor of 4561)
4561 / 3 = 1520.3333333333 (the remainder is 1, so 3 is not a divisor of 4561)
...
4561 / 66 = 69.106060606061 (the remainder is 7, so 66 is not a divisor of 4561)
4561 / 67 = 68.074626865672 (the remainder is 5, so 67 is not a divisor of 4561)