What are the divisors of 4641?

1, 3, 7, 13, 17, 21, 39, 51, 91, 119, 221, 273, 357, 663, 1547, 4641

There is a total of 16 positive divisors.
The sum of these divisors is 8064.
The arithmetic mean is 504.

16 odd divisors

1, 3, 7, 13, 17, 21, 39, 51, 91, 119, 221, 273, 357, 663, 1547, 4641

How to compute the divisors of 4641?

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 4641 by each of the numbers from 1 to 4641 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)

4641 / 1 = 4641 (the remainder is 0, so 1 is a divisor of 4641)
4641 / 2 = 2320.5 (the remainder is 1, so 2 is not a divisor of 4641)
4641 / 3 = 1547 (the remainder is 0, so 3 is a divisor of 4641)
...
4641 / 4640 = 1.0002155172414 (the remainder is 1, so 4640 is not a divisor of 4641)
4641 / 4641 = 1 (the remainder is 0, so 4641 is a divisor of 4641)

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 4641 (i.e. 68.124885321004). 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 $\frac{N}{d} = D$ )

4641 / 1 = 4641 (the remainder is 0, so 1 and 4641 are divisors of 4641)
4641 / 2 = 2320.5 (the remainder is 1, so 2 is not a divisor of 4641)
4641 / 3 = 1547 (the remainder is 0, so 3 and 1547 are divisors of 4641)
...
4641 / 67 = 69.268656716418 (the remainder is 18, so 67 is not a divisor of 4641)
4641 / 68 = 68.25 (the remainder is 17, so 68 is not a divisor of 4641)