What are the divisors of 1621?

1, 1621

There is a total of 2 positive divisors.
The sum of these divisors is 1622.
The arithmetic mean is 811.

2 odd divisors

1, 1621

How to compute the divisors of 1621?

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 1621 by each of the numbers from 1 to 1621 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)

1621 / 1 = 1621 (the remainder is 0, so 1 is a divisor of 1621)
1621 / 2 = 810.5 (the remainder is 1, so 2 is not a divisor of 1621)
1621 / 3 = 540.33333333333 (the remainder is 1, so 3 is not a divisor of 1621)
...
1621 / 1620 = 1.0006172839506 (the remainder is 1, so 1620 is not a divisor of 1621)
1621 / 1621 = 1 (the remainder is 0, so 1621 is a divisor of 1621)

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 1621 (i.e. 40.261644278395). 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 $\frac{N}{d} = D$ )

1621 / 1 = 1621 (the remainder is 0, so 1 and 1621 are divisors of 1621)
1621 / 2 = 810.5 (the remainder is 1, so 2 is not a divisor of 1621)
1621 / 3 = 540.33333333333 (the remainder is 1, so 3 is not a divisor of 1621)
...
1621 / 39 = 41.564102564103 (the remainder is 22, so 39 is not a divisor of 1621)
1621 / 40 = 40.525 (the remainder is 21, so 40 is not a divisor of 1621)