a-b-c-R-Find-triple-of-positive-real-numbers-a-b-c-that-satisfy-a-b-5-b-c-5-c-a-12-

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!!!