Question

Express H.C.F of 196 and 38220 in the form of 196 x+38220 y.

Answer 1

HCF (38220,  196) :
       38220 = 196 *  195 + 0            =>  HCF is  196.

      38220 =  196 *  194 + 196
   =>  196 = 38220 * 1 - 196 * 194
   =>  196  = 38220  * y + 196 * x,        where  y = 1  and x = - 194

 Also,   196   =  196 * 1 + 38220 * 0       So,  x = 1  and  y = 0.

Also,  38220 = 196 * 196 - 196
       =>  196 =  196 * 196 - 38220 * 1
                    = 196 * x + 38220 * y        So  x = 196  and  y = -1
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The normal way of finding the HCF and expressing the HCF as a linear combination of the given numbers is as follows:  This is an example to demonstrate the technique.

HCF (32 , 20) :
       32 =  20 * 1 + 12
       20 =  12 * 1 + 8
       12 =  8 * 1 + 4
         8 = 4 * 2  + 0        => 4 is the HCF.

  4 = 12 - 8 * 1
     = 12  -  (20 - 12 * 1 )
     = 2 * 12 - 20
     = 2 * (32 - 20 * 1 )  - 20
    4 = 2 * 32  - 3 * 20