Question

What will be the H.C.F of \[\left( {2 \times 3 \times 7 \times 9} \right)\] , \[\left( {2 \times 3 \times 9 \times 11} \right)\] and \[\left( {2 \times 3 \times 4 \times 5} \right)\] ?

Answer 1

Hint: H.C.F means Highest Common Factor. It is given by the product of least powers of common factors of the given numbers. If the given numbers do not have any common factor then, 1 will be its H.C.F as 1 is a factor of each and every number. All those numbers which divide a given number are its factors.

Complete step-by-step answer:
We are given three numbers, that are, \[\left( {2 \times 3 \times 7 \times 9} \right)\], \[\left( {2 \times 3 \times 9 \times 11} \right)\] and \[\left( {2 \times 3 \times 4 \times 5} \right)\]. We have to find its Highest Common Factor.
The first number is \[\left( {2 \times 3 \times 7 \times 9} \right)\] , we know that \[2,3,7\] are prime numbers, therefore they will have only 2 factors, that is 1 and the number itself. While 9 is composite number, It has factors, \[1,3,9\] so the given number can be written as \[\left( {2 \times 3 \times 7 \times 3 \times 3} \right)\]
The second number is \[\left( {2 \times 3 \times 9 \times 11} \right)\], we know that \[2,3,11\] are prime numbers, therefore they will have only two factors, that is one and the number itself. While $9$ is a composite number. It has factors, \[1,3,9\]. So, the given number can be written as \[\left( {2 \times 3 \times 3 \times 3 \times 11} \right)\]
The first number is \[\left( {2 \times 3 \times 4 \times 5} \right)\] , we know that \[2,3,5\] are prime numbers, therefore they will have only two factors, that is 1 and the number itself. While $4$ is a composite number, It has factors, \[1,2,4\]. So the given number can be written as \[\left( {2 \times 3 \times 2 \times 2 \times 5} \right)\].
Now, we know that H.C.F of given numbers is given by the product of least powers of common factors of the given numbers.
Here, the common factors are $2$ and $3$. Their least power is one.
So, H.C.F of given numbers is \[2 \times 3 = 6\].

Note: We should keep in mind the basic definition of H.C.F that it is given by the product of least powers of common factors of the given numbers. Also the factors are the numbers which divide the given numbers completely. The prime numbers have 2 factors only, that are, 1 and the given number itself.