What are the divisors of 4961?

1, 11, 41, 121, 451, 4961

There is a total of 6 positive divisors.
The sum of these divisors is 5586.
The arithmetic mean is 931.

6 odd divisors

1, 11, 41, 121, 451, 4961

How to compute the divisors of 4961?

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 4961 by each of the numbers from 1 to 4961 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)

4961 / 1 = 4961 (the remainder is 0, so 1 is a divisor of 4961)
4961 / 2 = 2480.5 (the remainder is 1, so 2 is not a divisor of 4961)
4961 / 3 = 1653.6666666667 (the remainder is 2, so 3 is not a divisor of 4961)
...
4961 / 4960 = 1.0002016129032 (the remainder is 1, so 4960 is not a divisor of 4961)
4961 / 4961 = 1 (the remainder is 0, so 4961 is a divisor of 4961)

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 4961 (i.e. 70.434366611761). 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 $\frac{N}{d} = D$ )

4961 / 1 = 4961 (the remainder is 0, so 1 and 4961 are divisors of 4961)
4961 / 2 = 2480.5 (the remainder is 1, so 2 is not a divisor of 4961)
4961 / 3 = 1653.6666666667 (the remainder is 2, so 3 is not a divisor of 4961)
...
4961 / 69 = 71.898550724638 (the remainder is 62, so 69 is not a divisor of 4961)
4961 / 70 = 70.871428571429 (the remainder is 61, so 70 is not a divisor of 4961)