Question

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

Answer 1

Hint: HCF of 65 and 117 can be written in the form of $ 65m - 117 $ .

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 \times 13 $

Prime factorization of $ 117 = 3 \times 3 \times 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 $ 65m - 117 $

$\Rightarrow 65m - 117 = 13 $

$\Rightarrow m = \dfrac{{130}}{{65}} = 2 $

Hence, the value of m is $ 2. $ $ $

Prime factorization of $ 65 = 5 \times 13 $

Prime factorization of $ 117 = 3 \times 3 \times 13 $

Here, $ 13 $ is common in the prime factors of both the numbers, hence, it would be considered one time in calculating LCM.

$\Rightarrow$ LCM = $ 5 \times 13 \times 3 \times 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.