Question in fraction (not simple )I have a question and its answer but I don't know how can i solve 37 13 = 2 + 1 x + 1 5 + 1 y the answer x = 1 , y = 2Could any one explain how to solve this ?? please

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Question in fraction (not simple )I have a question and its answer but I don&#039;t know how can i solve      37    13    =  2  +      1          x      +              1                  5          +                      1            y                              the answer   x  =  1  ,  y  =  2Could any one explain how to solve this ?? please

scoseBexgofvc · Accepted Answer

This is a development in continued fraction.At each step, you have to compute a quotient and remainder, like this:      37    13   has quotient   2 and remainder   11 , thus   37  =  13  ×  2  +  11  or also       37    13    =  2  +      11    13    =  2  +            1                        13          11                     Now you do the same with       13    11    =  1  +      2    11    thus       37    13    =  2  +            1              1        +                              2            11                                =  2  +            1              1        +                              1                                          11                2                                                        And finally,             11      2        =  5  +            1      2       , thus       37    13    =  2  +            1              1        +                              1                          5              +                                                1                  2

Averi Mitchell · Accepted Answer

37          13                                    =        2        +                  1                      x            +                          1                              5                +                                  1                  y                                                                                                  11          13                                    =                  1                      x            +                          1                              5                +                                  1                  y                                                                                                  13          11                                    =        x        +                  1                      5            +                          1              y                                                                                13            −            11            x                    11                                    =                  1                      5            +                          1              y                                                                    11                      13            −            11            x                                              =        5        +                  1          y                                                          −            54            +            55            x                                13            −            11            x                                              =                  1          y                                                          13            −            11            x                                −            54            +            55            x                                              =        y            Assuming x and y must be integers, note that  gcd  (  13  −  11  x  ,  −  54  +  55  x  )  =  gcd  (  13  −  11  x  ,  11  )  =  gcd  (  13  ,  11  )  =  1Hence  −  54  +  55  x  =  ±  1    ⟹    x  =  1Then  y  =            13      −      11              −      54      +      55        =  2.

Question in fraction (not simple ) I have a question and its answer but I don't know how can i solv

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