What are the divisors of 633?

1, 3, 211, 633

There is a total of 4 positive divisors.
The sum of these divisors is 848.
The arithmetic mean is 212.

4 odd divisors

1, 3, 211, 633

How to compute the divisors of 633?

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 633 by each of the numbers from 1 to 633 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)

633 / 1 = 633 (the remainder is 0, so 1 is a divisor of 633)
633 / 2 = 316.5 (the remainder is 1, so 2 is not a divisor of 633)
633 / 3 = 211 (the remainder is 0, so 3 is a divisor of 633)
...
633 / 632 = 1.001582278481 (the remainder is 1, so 632 is not a divisor of 633)
633 / 633 = 1 (the remainder is 0, so 633 is a divisor of 633)

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 633 (i.e. 25.159491250818). 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 $\frac{N}{d} = D$ )

633 / 1 = 633 (the remainder is 0, so 1 and 633 are divisors of 633)
633 / 2 = 316.5 (the remainder is 1, so 2 is not a divisor of 633)
633 / 3 = 211 (the remainder is 0, so 3 and 211 are divisors of 633)
...
633 / 24 = 26.375 (the remainder is 9, so 24 is not a divisor of 633)
633 / 25 = 25.32 (the remainder is 8, so 25 is not a divisor of 633)