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The HCF of 65 and 117 is expressible in the form 65m117 . Find the value of m. Also, find the LCM for 65 and 117 using the prime factorization method.

Answer
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Hint: HCF of 65 and 117 can be written in the form of 65m117 .
We have to find the factors of given two numbers using the prime factorization method.

Complete step-by-step answer:
Prime factorization of 65=5×13
Prime factorization of 117=3×3×13
Highest common factor means, the highest factor by which both the given numbers are completely divisible.
Here, the highest factor which is common among the two numbers is 13.
So, 65 and 117 are completely divisible by 13 .
Hence, the HCF of 65 and 117 is 13 .
And, The HCF of 65 and 117 is expressible in the form 65m117
65m117=13
m=13065=2
Hence, the value of m is 2.
Prime factorization of 65=5×13
Prime factorization of 117=3×3×13
Here, 13 is common in the prime factors of both the numbers, hence, it would be considered one time in calculating LCM.
LCM = 5×13×3×3=585
Hence, 585 is the LCM of 65 and117 which means that 585 is divisible by both 65 and 117 completely. 

Note: While solving this kind of problem, it is easy to first calculate the prime factors of the given numbers by the prime factorization method and then solve for LCM and HCF. 
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