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How do you find the Greatest Common Factor of 54, 27?

seo-qna
Last updated date: 13th Jun 2024
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Answer
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Hint: Here, we will find the Greatest Common Factors for the given numbers. We will find the factors of all the numbers separately by the method of prime factorization and by using the factors we will find the GCF of all the given numbers. Thus, the Greatest Common Factor of the numbers is the required answer.

Complete Step by Step Solution:
We are given with the numbers 54 and 27
Now, we will find the factors of all the numbers using the method of prime factorization
We will first do the prime factorization of 54.
We can see that 54 is an even number, so dividing it by the least prime number 2, we get
\[54 \div 2 = 27\]
Now dividing 27 by the next least prime number 3, we get
\[27 \div 3 = 9\]
Dividing 9 by 3, we get
\[9 \div 3 = 3\]
Here, the quotient is a prime number, so we will not divide the number further.
Thus, we can write 54 as \[54 = 2 \times 3 \times 3 \times 3\]
We will first do the prime factorization of 54.
Now, 27 is an odd number, so dividing it by the least odd prime number 3, we get
\[27 \div 3 = 9\]
Dividing 9 by 3, we get
\[9 \div 3 = 3\]
Here, the quotient is a prime number, so we will not divide the number further.
Thus, 27 can be written as \[27 = 3 \times 3 \times 3\]
Thus the factors of all the numbers are represented with the same bases as
\[\begin{array}{l}54 = {2^1} \times {3^3}\\27 = {2^0} \times {3^3}\end{array}\]
We will be using the method of prime factorization to find the Greatest Common Factor for all the given numbers.
Greatest common factor is a factor which is common for all the factors.
Thus, we get
\[GCF\left( {54,27} \right) = {3^3}\]
Applying the exponent on the terms, we get
\[ \Rightarrow GCF\left( {54,27} \right) = 27\]

Therefore, the GCF of 54 and 27 is 27.

Note:
We know that Prime Factorization is a method used for finding the factors of the given numbers. The Greatest Common Factor (G.C.F) of two numbers is defined as the greatest number which divides exactly both the numbers. The Least Common Multiple (L.C.M) of two numbers is defined as the smallest number which is divisible by both the numbers. GCF can be found by multiplying the factors with the least exponent common for all the factors and LCM can be found by multiplying the factors with the highest exponent from all the factors.