What are the divisors of 1327?

1, 1327

There is a total of 2 positive divisors.
The sum of these divisors is 1328.
The arithmetic mean is 664.

2 odd divisors

1, 1327

How to compute the divisors of 1327?

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 1327 by each of the numbers from 1 to 1327 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)

1327 / 1 = 1327 (the remainder is 0, so 1 is a divisor of 1327)
1327 / 2 = 663.5 (the remainder is 1, so 2 is not a divisor of 1327)
1327 / 3 = 442.33333333333 (the remainder is 1, so 3 is not a divisor of 1327)
...
1327 / 1326 = 1.000754147813 (the remainder is 1, so 1326 is not a divisor of 1327)
1327 / 1327 = 1 (the remainder is 0, so 1327 is a divisor of 1327)

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 1327 (i.e. 36.428011200174). 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 $\frac{N}{d} = D$ )

1327 / 1 = 1327 (the remainder is 0, so 1 and 1327 are divisors of 1327)
1327 / 2 = 663.5 (the remainder is 1, so 2 is not a divisor of 1327)
1327 / 3 = 442.33333333333 (the remainder is 1, so 3 is not a divisor of 1327)
...
1327 / 35 = 37.914285714286 (the remainder is 32, so 35 is not a divisor of 1327)
1327 / 36 = 36.861111111111 (the remainder is 31, so 36 is not a divisor of 1327)