What are the divisors of 4723?

1, 4723

There is a total of 2 positive divisors.
The sum of these divisors is 4724.
The arithmetic mean is 2362.

2 odd divisors

1, 4723

How to compute the divisors of 4723?

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 4723 by each of the numbers from 1 to 4723 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)

4723 / 1 = 4723 (the remainder is 0, so 1 is a divisor of 4723)
4723 / 2 = 2361.5 (the remainder is 1, so 2 is not a divisor of 4723)
4723 / 3 = 1574.3333333333 (the remainder is 1, so 3 is not a divisor of 4723)
...
4723 / 4722 = 1.0002117746717 (the remainder is 1, so 4722 is not a divisor of 4723)
4723 / 4723 = 1 (the remainder is 0, so 4723 is a divisor of 4723)

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 4723 (i.e. 68.724086025207). 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 $\frac{N}{d} = D$ )

4723 / 1 = 4723 (the remainder is 0, so 1 and 4723 are divisors of 4723)
4723 / 2 = 2361.5 (the remainder is 1, so 2 is not a divisor of 4723)
4723 / 3 = 1574.3333333333 (the remainder is 1, so 3 is not a divisor of 4723)
...
4723 / 67 = 70.492537313433 (the remainder is 33, so 67 is not a divisor of 4723)
4723 / 68 = 69.455882352941 (the remainder is 31, so 68 is not a divisor of 4723)