What are the divisors of 4577?

1, 23, 199, 4577

There is a total of 4 positive divisors.
The sum of these divisors is 4800.
The arithmetic mean is 1200.

4 odd divisors

1, 23, 199, 4577

How to compute the divisors of 4577?

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 4577 by each of the numbers from 1 to 4577 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)

4577 / 1 = 4577 (the remainder is 0, so 1 is a divisor of 4577)
4577 / 2 = 2288.5 (the remainder is 1, so 2 is not a divisor of 4577)
4577 / 3 = 1525.6666666667 (the remainder is 2, so 3 is not a divisor of 4577)
...
4577 / 4576 = 1.0002185314685 (the remainder is 1, so 4576 is not a divisor of 4577)
4577 / 4577 = 1 (the remainder is 0, so 4577 is a divisor of 4577)

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 4577 (i.e. 67.653529102331). 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 $\frac{N}{d} = D$ )

4577 / 1 = 4577 (the remainder is 0, so 1 and 4577 are divisors of 4577)
4577 / 2 = 2288.5 (the remainder is 1, so 2 is not a divisor of 4577)
4577 / 3 = 1525.6666666667 (the remainder is 2, so 3 is not a divisor of 4577)
...
4577 / 66 = 69.348484848485 (the remainder is 23, so 66 is not a divisor of 4577)
4577 / 67 = 68.313432835821 (the remainder is 21, so 67 is not a divisor of 4577)