What are the divisors of 1321?

1, 1321

There is a total of 2 positive divisors.
The sum of these divisors is 1322.
The arithmetic mean is 661.

2 odd divisors

1, 1321

How to compute the divisors of 1321?

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 1321 by each of the numbers from 1 to 1321 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)

1321 / 1 = 1321 (the remainder is 0, so 1 is a divisor of 1321)
1321 / 2 = 660.5 (the remainder is 1, so 2 is not a divisor of 1321)
1321 / 3 = 440.33333333333 (the remainder is 1, so 3 is not a divisor of 1321)
...
1321 / 1320 = 1.0007575757576 (the remainder is 1, so 1320 is not a divisor of 1321)
1321 / 1321 = 1 (the remainder is 0, so 1321 is a divisor of 1321)

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 1321 (i.e. 36.345563690772). 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 $\frac{N}{d} = D$ )

1321 / 1 = 1321 (the remainder is 0, so 1 and 1321 are divisors of 1321)
1321 / 2 = 660.5 (the remainder is 1, so 2 is not a divisor of 1321)
1321 / 3 = 440.33333333333 (the remainder is 1, so 3 is not a divisor of 1321)
...
1321 / 35 = 37.742857142857 (the remainder is 26, so 35 is not a divisor of 1321)
1321 / 36 = 36.694444444444 (the remainder is 25, so 36 is not a divisor of 1321)