If-GCD-a-b-1-and-GCD-c-d-1-a-b-c-d-1-a-lt-b-c-lt-d-is-it-possible-that-a-b-c-d-is-an-integer-number-

Question Number 26410 by Joel578 last updated on 25/Dec/17

If G C D (a, b) = 1 and G C D (c, d) = 1

a \neq b \neq c \neq d \neq 1, a < b, c < d

is it possible that \frac{a}{b} + \frac{c}{d} is an integer number ?

Answered by mrW1 last updated on 25/Dec/17

s i n c e g c d (c, d) = 1, \frac{c}{d} i s a l r e a d y r e d u c e d

t o l o w e s t t e r m s, i . e . \frac{c}{d} c a n n o t b e

s i m p l i f i e d f u r t h e r .

{l e t}^{'} s a s u m e \frac{a}{b} + \frac{c}{d} = n, t h e n

\frac{c}{d} = n - \frac{a}{b} = \frac{n b - a}{b} = \frac{e}{b} w i t h e = n b - a = i n t e g e r

t h a t m e a n s \frac{c}{d} c a n b e s i m p l i f i e d t o

\frac{e}{b} w i t h b < d . b u t t h i s i s a c o n t r a d i c t i o n .

\Rightarrow \frac{a}{b} + \frac{c}{d} c a n n o t b e a n i n t e g e r n u m b e r .

Commented by Joel578 last updated on 27/Dec/17

t h a n k y o u v e r y m u c h

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