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Find the HCF of 65 and 117 and express it in the form of 65m + 117n.

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Hint: First find the HCF of the given two numbers and use Euclid's Division Lemma method to find the equation of the given form.

 

Complete step-by-step answer:

As we know that according to Euclid's Division Lemma any number can be written as,

   $\Rightarrow a = bq + r $...........................(1)

  Where ‘a’ is dividend ‘b’ is divisor, q is quotient and r is remainder

 And as we see here $ 117 > 65 $

 So, writing 117 in the form of equation 1 we get

 $  \Rightarrow 117 = 65*1 + 52$.............................(2)

 Now 65 will be dividend and 52 will be divisor

 $  \Rightarrow 65 = 52*1 + 13$................................(3)

 Now 52 will be dividend and 13 will be divisor

  $ \Rightarrow 52 = 13*4 + 0 $

  As now the remainder is 0. So, HCF will be the last divisor.

  So, HCF of 117 and 65 will be 13.

  According to the question,

   $\Rightarrow 13 = 65m + 117n $..................................(4)

  So, we need to find value of m and n 

  From equation 3 we get,

   $\Rightarrow 13 = 65 - 52*1 $........................................(5)

  From equation 2 we get,

  $ \Rightarrow 52 = 117 - 65*1 $....................................(6)

 Putting value of from equation 6 to equation 5 we will get,

  $ \Rightarrow 13 = 65 - \left( {117 - 65*1} \right)$.........................(7)

  Solving equation 7 we get,

  $ \Rightarrow 13 = 62*(2) + 117*( - 1)$.......................................(8)

  On comparing equation 8 and 4 we get value of m = 2 and n = - 1 

  Now HCF of 65 and 117 in form of 65m + 117n is,

 $  \Rightarrow 13 = 62*(2) + 117*( - 1){\text{ }} $

 

NOTE: - Whenever you come up with this type of problem then the best way is to calculate HCF and then solve all equations for calculating unknown variables. According to Euclid's Division Lemma any number can be written in the form of $a = bq + r$ where $a > b$.


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