a, b, c ∈ R^+ Find triple of positive real numbers (a, b, c) that satisfy a⌊b⌋ = 5 b⌊c⌋ = 5 c⌊a⌋ = 12


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Question Number 58700 by naka3546 last updated on 28/Apr/19

$a, b, c \in R^{+}$ $F i n d t r i p l e o f p o s i t i v e r e a l n u m b e r s (a, b, c) t h a t s a t i s f y$ $a ⌊ b ⌋ = 5$ $b ⌊ c ⌋ = 5$ $c ⌊ a ⌋ = 12$
Commented by Rasheed.Sindhi last updated on 28/Apr/19

$I n a ⌊ b ⌋ = 5, ⌊ b ⌋ i s a n i n t e g e r$ $H e n c e a i s a l s o a n i n t e g e r$ $r e a l \times i n t e g e r = i n t e g e r \Rightarrow r e a l = i n t e g e r .$ $S o a, b, c \in Z$ $⌊ a ⌋ = a, ⌊ b ⌋ = b, ⌊ c ⌋ = c$ $a b = 5$ $b c = 5$ $c a = 12$ $a^{2} b^{2} c^{2} = 5 \times 5 \times 12 = 300 = 10 \sqrt{3} ? ? ? ?$ $B u t,$ $a, b, c \in Z \Rightarrow a^{2}, b^{2}, c^{2} \in Z \Rightarrow a^{2} b^{2} c^{2} \in Z$ $T h i s c o n t r a d i c t i o n l e a d s t h a t t h e r e$ ${a r e n}^{'} t a n y s u c h a, b, c .$
Commented by mr W last updated on 28/Apr/19

$d e a r s i r, a \times ⌊ b ⌋ = i n t e g e r {d o e s n}^{'} t m e a n$ $t h a t a i s i n t e g e r . i t m e a n s o n l y t h a t$ $a i s r a t i o n a l . e . g . a = \frac{5}{3}, b = \frac{7}{2}, ⌊ b ⌋ = 3,$ $\Rightarrow a \times ⌊ b ⌋ = 5 .$
Commented by Rasheed.Sindhi last updated on 28/Apr/19

$S o r r y f o r d e f f e c t i v e l o g i c!$ ${Y o u}^{'} r e v e r y r i g h t s i r! T h a n k y o u$ $t o c o r r e c t m e . A c t u a l l y I^{'} m h e r e t o l e a r n$ $f r o m y o u!$
Commented by mr W last updated on 28/Apr/19

${Y o u}^{'} r e w e l c o m e s i r . I n f a c t I h a v e$ $l e a r n t a n d s t i l l l e a r n a l o t f r o m y o u s i r .$ $M a n y p o s t s o f y o u h a v e b e c o m e c l a s s i c$ $o f t h e f o r u m . {I t}^{'} s a p l e a s u r e t o s t u d y$ $t h e m .$
Commented by Rasheed.Sindhi last updated on 29/Apr/19

$TH α n k s t o e n h e a r t e n m e i n s u c h a n$ $e x c i t i n g w a y s i r!!!$
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