What are the divisors of 1633?

1, 23, 71, 1633

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

4 odd divisors

1, 23, 71, 1633

How to compute the divisors of 1633?

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 1633 by each of the numbers from 1 to 1633 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)

1633 / 1 = 1633 (the remainder is 0, so 1 is a divisor of 1633)
1633 / 2 = 816.5 (the remainder is 1, so 2 is not a divisor of 1633)
1633 / 3 = 544.33333333333 (the remainder is 1, so 3 is not a divisor of 1633)
...
1633 / 1632 = 1.000612745098 (the remainder is 1, so 1632 is not a divisor of 1633)
1633 / 1633 = 1 (the remainder is 0, so 1633 is a divisor of 1633)

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 1633 (i.e. 40.410394702353). 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 $\frac{N}{d} = D$ )

1633 / 1 = 1633 (the remainder is 0, so 1 and 1633 are divisors of 1633)
1633 / 2 = 816.5 (the remainder is 1, so 2 is not a divisor of 1633)
1633 / 3 = 544.33333333333 (the remainder is 1, so 3 is not a divisor of 1633)
...
1633 / 39 = 41.871794871795 (the remainder is 34, so 39 is not a divisor of 1633)
1633 / 40 = 40.825 (the remainder is 33, so 40 is not a divisor of 1633)