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The HCF of ${3^5}$, ${3^9}$ and ${3^{14}}$ is:
(A) ${3^5}$
(B) ${3^9}$
(C) ${3^{14}}$
(D) ${3^{21}}$

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Last updated date: 22nd Jul 2024
Total views: 351.6k
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Answer
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Hint: There are various methods for finding the highest common factor of the given numbers. The simplest method is to find the highest common factor is by prime factorization method. In the prime factorization method, we first represent the given two numbers as a product of their prime factors and then find the product of the lowest powers of all the common factors.

Complete step by step solution:
In the given question, we are required to find the highest common factor of ${3^5}$, ${3^9}$ and ${3^{14}}$.
To find the highest common factor of ${3^5}$, ${3^9}$ and ${3^{14}}$, first we find out the prime factors of all the numbers.
Prime factors of ${3^5}$$ = 3 \times 3 \times 3 \times 3 \times 3$
Prime factors of ${3^9}$$ = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
Prime factors of ${3^{14}}$$ = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
Now, the highest common factor is the product of the lowest powers of all the common factors.
Now, we can see that there is only one repeated factor,$3$ in all the numbers.
Hence, least common multiple of ${3^5}$, ${3^9}$ and ${3^{14}}$$ = 3 \times 3 \times 3 \times 3 \times 3$
$ = {3^5}$
Hence, the highest common factor of ${3^5}$, ${3^9}$ and ${3^{14}}$ is ${3^5}$.

Note:
Highest common factor is the greatest number that divides both the given numbers. Similarly, the highest common factor can also be found by using the prime factorization method as well as using Euclid’s division lemma. Highest common divisor is just a product of common factors with lowest power. Highest common factor can also be calculated by division method as well as by using Euclid’s division lemma.