What are the divisors of 1643?

1, 31, 53, 1643

There is a total of 4 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 432.

4 odd divisors

1, 31, 53, 1643

How to compute the divisors of 1643?

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 1643 by each of the numbers from 1 to 1643 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)

1643 / 1 = 1643 (the remainder is 0, so 1 is a divisor of 1643)
1643 / 2 = 821.5 (the remainder is 1, so 2 is not a divisor of 1643)
1643 / 3 = 547.66666666667 (the remainder is 2, so 3 is not a divisor of 1643)
...
1643 / 1642 = 1.0006090133983 (the remainder is 1, so 1642 is not a divisor of 1643)
1643 / 1643 = 1 (the remainder is 0, so 1643 is a divisor of 1643)

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 1643 (i.e. 40.533936399022). 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 $\frac{N}{d} = D$ )

1643 / 1 = 1643 (the remainder is 0, so 1 and 1643 are divisors of 1643)
1643 / 2 = 821.5 (the remainder is 1, so 2 is not a divisor of 1643)
1643 / 3 = 547.66666666667 (the remainder is 2, so 3 is not a divisor of 1643)
...
1643 / 39 = 42.128205128205 (the remainder is 5, so 39 is not a divisor of 1643)
1643 / 40 = 41.075 (the remainder is 3, so 40 is not a divisor of 1643)