Question

Find the H.C.F of the following:
18, 54, 81

Answer 1

Hint: To find the H.C.F of the three numbers (18, 54, 81), we need to first of all find the prime factorization of the three numbers separately. Then, we are going to find the prime factors which are common in all the three numbers. After finding the common prime factors, we are going to multiply the prime factors and the result of the multiplication of the prime factors is the H.C.F of the three numbers.

Complete step by step answer:
The three numbers given in the above problem of which we have to find the H.C.F are as follows:
18, 54, 81
Now, we are going to write the prime factorization of these three numbers one by one.
First of all, we are going to write the prime factorization for 18 which is equal to:
$18=2\times 3\times 3$
Now, we are going to write the prime factorization for 54 which is equal to:
$54=2\times 3\times 3\times 3$
Now, we are going to write the prime factorization for 81 which is equal to:
$81=3\times 3\times 3\times 3$
After that we are going to find the common factors among the three numbers by writing all of three prime factorizations first then underline the common factors among the three numbers:
$18=2\times \underline{3\times 3}$
$54=2\times 3\times \underline{3\times 3}$
$81=3\times 3\times \underline{3\times 3}$
As you can see that $3\times 3$ is the common factor amongst the three numbers which we have shown by underlining them.
Multiplying the two factors $3\times 3$ we get 9.

Hence, the H.C.F of the three numbers (18, 54, 81) is 9.

Note: The mistake that could be possible in the above problem is that while finding the common factors amongst the three numbers you might mistakenly not consider that the common factors must be common in the three numbers not two of them have common factors but the third one does not contain those common factors.
For e.g. in the above prime factorization of the three numbers:
$18=2\times 3\times 3$
$54=2\times 3\times 3\times 3$
$81=3\times 3\times 3\times 3$
Now, if we consider 18 and 54 then the common factors in these two numbers are: $2\times 3\times 3$ but 2 is not the common factor in 81 so this is the mistake we were discussing in the above paragraph.