What are the divisors of 9601?

1, 9601

There is a total of 2 positive divisors.
The sum of these divisors is 9602.
The arithmetic mean is 4801.

2 odd divisors

1, 9601

How to compute the divisors of 9601?

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 9601 by each of the numbers from 1 to 9601 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)

9601 / 1 = 9601 (the remainder is 0, so 1 is a divisor of 9601)
9601 / 2 = 4800.5 (the remainder is 1, so 2 is not a divisor of 9601)
9601 / 3 = 3200.3333333333 (the remainder is 1, so 3 is not a divisor of 9601)
...
9601 / 9600 = 1.0001041666667 (the remainder is 1, so 9600 is not a divisor of 9601)
9601 / 9601 = 1 (the remainder is 0, so 9601 is a divisor of 9601)

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 9601 (i.e. 97.984692682072). 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 $\frac{N}{d} = D$ )

9601 / 1 = 9601 (the remainder is 0, so 1 and 9601 are divisors of 9601)
9601 / 2 = 4800.5 (the remainder is 1, so 2 is not a divisor of 9601)
9601 / 3 = 3200.3333333333 (the remainder is 1, so 3 is not a divisor of 9601)
...
9601 / 96 = 100.01041666667 (the remainder is 1, so 96 is not a divisor of 9601)
9601 / 97 = 98.979381443299 (the remainder is 95, so 97 is not a divisor of 9601)