What are the divisors of 1663?

1, 1663

There is a total of 2 positive divisors.
The sum of these divisors is 1664.
The arithmetic mean is 832.

2 odd divisors

1, 1663

How to compute the divisors of 1663?

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 1663 by each of the numbers from 1 to 1663 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)

1663 / 1 = 1663 (the remainder is 0, so 1 is a divisor of 1663)
1663 / 2 = 831.5 (the remainder is 1, so 2 is not a divisor of 1663)
1663 / 3 = 554.33333333333 (the remainder is 1, so 3 is not a divisor of 1663)
...
1663 / 1662 = 1.0006016847172 (the remainder is 1, so 1662 is not a divisor of 1663)
1663 / 1663 = 1 (the remainder is 0, so 1663 is a divisor of 1663)

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 1663 (i.e. 40.779897008207). 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 $\frac{N}{d} = D$ )

1663 / 1 = 1663 (the remainder is 0, so 1 and 1663 are divisors of 1663)
1663 / 2 = 831.5 (the remainder is 1, so 2 is not a divisor of 1663)
1663 / 3 = 554.33333333333 (the remainder is 1, so 3 is not a divisor of 1663)
...
1663 / 39 = 42.641025641026 (the remainder is 25, so 39 is not a divisor of 1663)
1663 / 40 = 41.575 (the remainder is 23, so 40 is not a divisor of 1663)