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Find the largest number which is a factor of $ 180 $ and $ 336 $ .

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Last updated date: 20th Jul 2024
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Answer
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Hint: In the given problem, we are required to find the largest number which is the factor of the given two numbers. Basically, we have to find the highest common factor of the two numbers provided to us in the problem itself. There are various methods for finding the highest common factor of the given numbers. The simplest method 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 in order to compute the HCF of the given numbers.

Complete step-by-step answer:
In the given question, we are required to find the largest number which is a factor of $ 180 $ and $ 336 $ .
So, we have to evaluate the HCF of the numbers $ 180 $ and $ 336 $ .
To find the highest common factor of $ 180 $ and $ 336 $ , first we find the prime factors of both the numbers. So, we get,
Prime factorization of $ 180 $ $ = 3 \times 3 \times 2 \times 2 \times 5 $
 $ = {3^2} \times {2^2} \times 5 $
Prime factorization of $ 336 $ $ = 3 \times 2 \times 2 \times 2 \times 2 \times 7 $
 $ = 3 \times {2^4} \times 7 $
Now, the highest common factor is the product of the lowest powers of all the common factors.
Now, we can see that $ 2 $ and $ 3 $ are the common factors in the numbers $ 180 $ and $ 336 $ .
So, HCF of $ 180 $ and $ 336 $ $ = 3 \times {2^2} = 12 $
Hence, the highest common factor of $ 180 $ and $ 336 $ is $ 12 $ .
Hence, the largest number which is a factor of $ 180 $ and $ 336 $ is $ 12 $ .
So, the correct answer is “12”.

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 of two numbers is also the largest number that is a factor of both the numbers.