Question

Find the H.C.F of the following numbers:
30, 42

Answer 1

Hint: To find the H.C.F of the two numbers (30, 42), we need to first of all find the prime factorization of the two numbers separately. Then, we are going to find the prime factors which are common in all the two 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 two numbers.

Complete step-by-step answer:
The two numbers given in the above problem of which we have to find the H.C.F are as follows:
30, 42
Now, we are going to write the prime factorization of these two numbers one by one.
First of all, we are going to write the prime factorization for 30 which is equal to:
$30=2\times 3\times 5$
Now, we are going to write the prime factorization for 42 which is equal to:
$42=2\times 3\times 7$
After that we are going to find the common factors among the two numbers by writing all of two prime factorizations first then underline the common factors among the two numbers:
$30=\underset{―}{2\times 3}\times 5$
$42=\underset{―}{2\times 3}\times 7$
As you can see that $2\times 3$ is the common factor amongst the two numbers which we have shown by underlining them.
Multiplying the two factors $2\times 3$ we get 6.
Hence, the H.C.F of the two numbers (30, 42) is 6.

Note: Using H.C.F, we can find the L.C.M of the two numbers also and not only two numbers three, four any number we can find. So, the trick is we have to multiply the H.C.F of the two numbers with the non-common numbers from each of 30 and 42. The numbers which are not common in 30 and 42 are: 5 and 7 which you can see from the above prime factorization.
$30=\underset{―}{2\times 3}\times 5$
$42=\underset{―}{2\times 3}\times 7$
In the above, the non-underlined numbers are the non-common numbers.
So, to find the L.C.M we are going to multiply H.C.F which is equal to 6 to the non common numbers (5 and 7) we get,
$6\times 5\times 7=210$