What are the common factor of 63 and 135?

We divide 63 and 135 into their multiples to find all of their factors. Let's use 63 as an example. It can be divided by 1 to equal 63, which are our first two factors, ${\displaystyle \{1,63\}}$.
The next thing we notice is that 63 can be divided by three to get 21, which are the next two factors, giving us ${\displaystyle 1,3,21,63}$.
Finally, we discover that the last two factors of ${\displaystyle \{1,3,7,9,21,63\}}$ can be calculated by dividing 63 by 7, which results in 9. Since there are no more pairs of integers that, when multiplied, equal 63, these are all the factors of 63.
We then do the same with 135 to find its factor list is ${\displaystyle \{1,3,5,9,15,27,45,135\}}$. Finally, we see which elements are present in both sets, which happen to be {1,3,9}.
The highest integer that divides into two or more numbers to produce a third integer is known as the highest common factor, or HCF. The HCF can be acquired in two different ways. The first method involves manually locating all of the 63 factors (1,3,7,9,21,63) and all of the 135 factors (1,3,5,9,15,27,45,135) and comparing them to determine that their HCF is 9.
The second way is by dividing both numbers ${\displaystyle =\frac{135}{63}}$, simplifying the fraction ${\displaystyle =\frac{15}{7}}$, then dividing the starting number with the new simplifiied number,
${\displaystyle \frac{135}{15}=9}$ or ${\displaystyle \frac{63}{7}=9}$,
Keeping in mind to always divide the denominator by the denominator and the numerator by the numerator.
This process works with any two numbers which you want to find the HCF of, and can be simplified into this rule:
If a= any number, b= any number, and ${\displaystyle \frac{c}{d}}$ is the simplified fraction of ${\displaystyle \frac{a}{b}}$,
The HCF ${\displaystyle =\frac{a}{c}}$ or ${\displaystyle =\frac{b}{d}}$.