What are the divisors of 637?

1, 7, 13, 49, 91, 637

There is a total of 6 positive divisors.
The sum of these divisors is 798.
The arithmetic mean is 133.

6 odd divisors

1, 7, 13, 49, 91, 637

How to compute the divisors of 637?

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 637 by each of the numbers from 1 to 637 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)

637 / 1 = 637 (the remainder is 0, so 1 is a divisor of 637)
637 / 2 = 318.5 (the remainder is 1, so 2 is not a divisor of 637)
637 / 3 = 212.33333333333 (the remainder is 1, so 3 is not a divisor of 637)
...
637 / 636 = 1.001572327044 (the remainder is 1, so 636 is not a divisor of 637)
637 / 637 = 1 (the remainder is 0, so 637 is a divisor of 637)

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 637 (i.e. 25.238858928248). 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 $\frac{N}{d} = D$ )

637 / 1 = 637 (the remainder is 0, so 1 and 637 are divisors of 637)
637 / 2 = 318.5 (the remainder is 1, so 2 is not a divisor of 637)
637 / 3 = 212.33333333333 (the remainder is 1, so 3 is not a divisor of 637)
...
637 / 24 = 26.541666666667 (the remainder is 13, so 24 is not a divisor of 637)
637 / 25 = 25.48 (the remainder is 12, so 25 is not a divisor of 637)