What are the divisors of 1627?

1, 1627

There is a total of 2 positive divisors.
The sum of these divisors is 1628.
The arithmetic mean is 814.

2 odd divisors

1, 1627

How to compute the divisors of 1627?

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 1627 by each of the numbers from 1 to 1627 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)

1627 / 1 = 1627 (the remainder is 0, so 1 is a divisor of 1627)
1627 / 2 = 813.5 (the remainder is 1, so 2 is not a divisor of 1627)
1627 / 3 = 542.33333333333 (the remainder is 1, so 3 is not a divisor of 1627)
...
1627 / 1626 = 1.0006150061501 (the remainder is 1, so 1626 is not a divisor of 1627)
1627 / 1627 = 1 (the remainder is 0, so 1627 is a divisor of 1627)

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 1627 (i.e. 40.336088060197). 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 $\frac{N}{d} = D$ )

1627 / 1 = 1627 (the remainder is 0, so 1 and 1627 are divisors of 1627)
1627 / 2 = 813.5 (the remainder is 1, so 2 is not a divisor of 1627)
1627 / 3 = 542.33333333333 (the remainder is 1, so 3 is not a divisor of 1627)
...
1627 / 39 = 41.717948717949 (the remainder is 28, so 39 is not a divisor of 1627)
1627 / 40 = 40.675 (the remainder is 27, so 40 is not a divisor of 1627)