What are the divisors of 1371?

1, 3, 457, 1371

There is a total of 4 positive divisors.
The sum of these divisors is 1832.
The arithmetic mean is 458.

4 odd divisors

1, 3, 457, 1371

How to compute the divisors of 1371?

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 1371 by each of the numbers from 1 to 1371 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)

1371 / 1 = 1371 (the remainder is 0, so 1 is a divisor of 1371)
1371 / 2 = 685.5 (the remainder is 1, so 2 is not a divisor of 1371)
1371 / 3 = 457 (the remainder is 0, so 3 is a divisor of 1371)
...
1371 / 1370 = 1.0007299270073 (the remainder is 1, so 1370 is not a divisor of 1371)
1371 / 1371 = 1 (the remainder is 0, so 1371 is a divisor of 1371)

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 1371 (i.e. 37.027017163147). 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 $\frac{N}{d} = D$ )

1371 / 1 = 1371 (the remainder is 0, so 1 and 1371 are divisors of 1371)
1371 / 2 = 685.5 (the remainder is 1, so 2 is not a divisor of 1371)
1371 / 3 = 457 (the remainder is 0, so 3 and 457 are divisors of 1371)
...
1371 / 36 = 38.083333333333 (the remainder is 3, so 36 is not a divisor of 1371)
1371 / 37 = 37.054054054054 (the remainder is 2, so 37 is not a divisor of 1371)