Question

Find the H.C.F. of given numbers by prime factorization method: \[54,72,90\]

Answer 1

Hint: We have to H.C.F. of given numbers by prime factorization method to solve the problem.
The prime factorization method involves certain steps which are as given below:
We know that \[2\]is the first prime number. So, start dividing the given number by \[2\], continue the process of the division until you get the decimal or a remainder. Then divide by next prime number \[3\] and then divide by next prime numbers \[5,7\]and so on, until we remain with only prime numbers.

Complete step by step answer:
We need to find the H.C.F. of$54,72,90$ by prime factorization method.
Let us apply the Prime factorization algorithm to numbers$54$.
We know that \[2\]is the first prime number. So, start dividing the given number by\[2\]
On factoring by $2$ we get,
$54 = 2 \times 27$
Further on factoring by $3$ we get,
$54 = 2 \times 3 \times 9$
Further on factoring by $3$ we get,
$54 = \underline {2 \times 3 \times 3} \times 3$ \[......\left[ 1 \right]\]
Now all the numbers that remained are prime numbers. So this is the prime factorization of $54$.
Let us apply the Prime factorization algorithm to numbers $72$.
We know that \[2\]is the first prime number. So, start dividing the given number by \[2\]
On factorizing by $2$ we get,
$72 = 2 \times 36$
Further on factorizing by $2$ we get,
$72 = 2 \times 2 \times 18$
Further on factorizing by $2$ we get,
$72 = 2 \times 2 \times 2 \times 9$
Further on factorizing by $3$ we get,
$72 = 2 \times 2 \times \underline {2 \times 3 \times 3} $ \[......\left[ 2 \right]\]
Now all the numbers that remained are prime numbers. So this is the prime factorization $72$.
Let us apply the Prime factorization algorithm to numbers $90$.
We know that \[2\]is the first prime number. So, start dividing the given number by \[2\]
On factorizing by $2$ we get,
$90 = 2 \times 45$
Further on factorizing by $3$ we get,
$90 = 2 \times 3 \times 15$
Further on factorizing by $3$ we get,
$90 = \underline {2 \times 3 \times 3} \times 5$ \[......\left[ 3 \right]\]
Now all the numbers that remained are prime numbers. So this is the prime factorization of $90$.
Let us obtain H.C.F of $54,72,90$
On observing equations $1,2{\text{ and }}3$,
From the final prime factors of the numbers, $54,72,90$we can mark the common factors in all three numbers are \[2 \times 3 \times 3\]
Thus the H.C.F. of $54,72,90 = 2 \times 3 \times 3$
H.C.F. of $54,72,90 = 18$
This is the required H.C.F. value.

Note:
 Prime factorization algorithm is the simplest algorithm of factorization. In this method, any number is factorized into prime factors. The first few prime numbers are $2,3,7,11,13$etc. Prime factorization algorithm is very useful for calculating H.C.F. of multiple numbers.