What are the divisors of 4731?

1, 3, 19, 57, 83, 249, 1577, 4731

There is a total of 8 positive divisors.
The sum of these divisors is 6720.
The arithmetic mean is 840.

8 odd divisors

1, 3, 19, 57, 83, 249, 1577, 4731

How to compute the divisors of 4731?

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 4731 by each of the numbers from 1 to 4731 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)

4731 / 1 = 4731 (the remainder is 0, so 1 is a divisor of 4731)
4731 / 2 = 2365.5 (the remainder is 1, so 2 is not a divisor of 4731)
4731 / 3 = 1577 (the remainder is 0, so 3 is a divisor of 4731)
...
4731 / 4730 = 1.0002114164905 (the remainder is 1, so 4730 is not a divisor of 4731)
4731 / 4731 = 1 (the remainder is 0, so 4731 is a divisor of 4731)

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 4731 (i.e. 68.782265156071). 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 $\frac{N}{d} = D$ )

4731 / 1 = 4731 (the remainder is 0, so 1 and 4731 are divisors of 4731)
4731 / 2 = 2365.5 (the remainder is 1, so 2 is not a divisor of 4731)
4731 / 3 = 1577 (the remainder is 0, so 3 and 1577 are divisors of 4731)
...
4731 / 67 = 70.611940298507 (the remainder is 41, so 67 is not a divisor of 4731)
4731 / 68 = 69.573529411765 (the remainder is 39, so 68 is not a divisor of 4731)