What are the divisors of 4691?

1, 4691

There is a total of 2 positive divisors.
The sum of these divisors is 4692.
The arithmetic mean is 2346.

2 odd divisors

1, 4691

How to compute the divisors of 4691?

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 4691 by each of the numbers from 1 to 4691 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)

4691 / 1 = 4691 (the remainder is 0, so 1 is a divisor of 4691)
4691 / 2 = 2345.5 (the remainder is 1, so 2 is not a divisor of 4691)
4691 / 3 = 1563.6666666667 (the remainder is 2, so 3 is not a divisor of 4691)
...
4691 / 4690 = 1.0002132196162 (the remainder is 1, so 4690 is not a divisor of 4691)
4691 / 4691 = 1 (the remainder is 0, so 4691 is a divisor of 4691)

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 4691 (i.e. 68.490875304671). 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 $\frac{N}{d} = D$ )

4691 / 1 = 4691 (the remainder is 0, so 1 and 4691 are divisors of 4691)
4691 / 2 = 2345.5 (the remainder is 1, so 2 is not a divisor of 4691)
4691 / 3 = 1563.6666666667 (the remainder is 2, so 3 is not a divisor of 4691)
...
4691 / 67 = 70.014925373134 (the remainder is 1, so 67 is not a divisor of 4691)
4691 / 68 = 68.985294117647 (the remainder is 67, so 68 is not a divisor of 4691)