What are the divisors of 1301?

1, 1301

There is a total of 2 positive divisors.
The sum of these divisors is 1302.
The arithmetic mean is 651.

2 odd divisors

1, 1301

How to compute the divisors of 1301?

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 1301 by each of the numbers from 1 to 1301 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)

1301 / 1 = 1301 (the remainder is 0, so 1 is a divisor of 1301)
1301 / 2 = 650.5 (the remainder is 1, so 2 is not a divisor of 1301)
1301 / 3 = 433.66666666667 (the remainder is 2, so 3 is not a divisor of 1301)
...
1301 / 1300 = 1.0007692307692 (the remainder is 1, so 1300 is not a divisor of 1301)
1301 / 1301 = 1 (the remainder is 0, so 1301 is a divisor of 1301)

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 1301 (i.e. 36.069377593743). 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 $\frac{N}{d} = D$ )

1301 / 1 = 1301 (the remainder is 0, so 1 and 1301 are divisors of 1301)
1301 / 2 = 650.5 (the remainder is 1, so 2 is not a divisor of 1301)
1301 / 3 = 433.66666666667 (the remainder is 2, so 3 is not a divisor of 1301)
...
1301 / 35 = 37.171428571429 (the remainder is 6, so 35 is not a divisor of 1301)
1301 / 36 = 36.138888888889 (the remainder is 5, so 36 is not a divisor of 1301)