Question

What are all the greatest common factors of $36$ and $90$ ?

Answer 1

Hint: To find the greatest common factor of any two numbers, first write the numbers as the product of their primes. Then out of those primes take out the common factors in pairs and multiply them to get the greatest common factor.

Complete step by step solution:
The numbers given to us are $36$ and $90$.
To find their greatest common factor we have to first write them as the products of their primes, that is,
$36=2\times 2\times 3\times 3$ And $90=2\times 3\times 3\times 5$
When we find common factors of any two or more given numbers, then there can be several common factors but there will always be one greatest common factor.
A greatest common factor can be defined as that number which will divide all the numbers given without leaving a remainder.
So, for the above numbers $36$ and $90$, we can see that there is one pair of $2's$ and two pairs of $3's$ common in the factors of the given two numbers.
Therefore, the greatest common factor can be calculated to be,
$\Rightarrow G.C.F=2\times 3\times 3=18$
So, we can see that the greatest common factor of the given numbers is $18$ .
As, for all the common factors of both the numbers, it will be helpful to write down the all factors of both the numbers and then select from them to see which are common,
For example:
Factors of $36$ are $1,2,3,4,6,9,12,18$ and $36$ itself.
While factors of $90$ are $1,2,3,5,6,9,10,15,18,30,45$ and $90$ itself.
By comparing all the factors, we can see that the common factors for both the numbers are:
$1,2,3,6,9$ and $18$
Since, $18$ is the highest among all, it is also the greatest common factor.

Note: The above question can also be solved in an alternative way, by first making the list of all the common factors for the given numbers and then picking the largest or the greatest one that is common.
Always remember that there can be many common factors but there will be only one Greatest Common Factor or G.C.F.