What are the divisors of 4965?

1, 3, 5, 15, 331, 993, 1655, 4965

There is a total of 8 positive divisors.
The sum of these divisors is 7968.
The arithmetic mean is 996.

8 odd divisors

1, 3, 5, 15, 331, 993, 1655, 4965

How to compute the divisors of 4965?

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 4965 by each of the numbers from 1 to 4965 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)

4965 / 1 = 4965 (the remainder is 0, so 1 is a divisor of 4965)
4965 / 2 = 2482.5 (the remainder is 1, so 2 is not a divisor of 4965)
4965 / 3 = 1655 (the remainder is 0, so 3 is a divisor of 4965)
...
4965 / 4964 = 1.0002014504432 (the remainder is 1, so 4964 is not a divisor of 4965)
4965 / 4965 = 1 (the remainder is 0, so 4965 is a divisor of 4965)

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 4965 (i.e. 70.462756119811). 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 $\frac{N}{d} = D$ )

4965 / 1 = 4965 (the remainder is 0, so 1 and 4965 are divisors of 4965)
4965 / 2 = 2482.5 (the remainder is 1, so 2 is not a divisor of 4965)
4965 / 3 = 1655 (the remainder is 0, so 3 and 1655 are divisors of 4965)
...
4965 / 69 = 71.95652173913 (the remainder is 66, so 69 is not a divisor of 4965)
4965 / 70 = 70.928571428571 (the remainder is 65, so 70 is not a divisor of 4965)