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

Semaj Christian

Semaj Christian

Answered question

2022-06-13

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 = 2
Could any one explain how to solve this ?? please

Answer & Explanation

scoseBexgofvc

scoseBexgofvc

Beginner2022-06-14Added 20 answers

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

Averi Mitchell

Beginner2022-06-15Added 8 answers

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 ) = 1
Hence
54 + 55 x = ± 1 x = 1
Then
y = 13 11 54 + 55 = 2.

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