Find-out-x-y-such-that-gcd-x-3-y-2-gcd-x-2-y-3-

Question Number 62244 by Rasheed.Sindhi last updated on 18/Jun/19

Find out x, y, such that

\gcd (x^{3}, y^{2}) = \gcd (x^{2}, y^{3})

Commented by mr W last updated on 18/Jun/19

o n e s o l u t i o n i s :

x = Π p_{i}^{n_{i}}

y = Π q_{j}^{m_{j}}

w i t h p, q = p r i m e n u m b e r a n d p \neq q

Commented by Rasheed.Sindhi last updated on 19/Jun/19

T h a n k s S i r!

W a i t i n g f o r s o l u t i o n s w h e n x & y a r e

n o n c o p r i m e w i t h s o m e w o r k i n g s .

Commented by Rasheed.Sindhi last updated on 23/Jun/19

T h e q u e s t i o n h a v i n g n o m a r k e t v a l u e!

Answered by Rasheed.Sindhi last updated on 21/Jun/19

x = Π p_{i}^{a_{i}} y = Π p_{i}^{b_{i}} : p_{i} \in P \land a_{i}, b_{i} \in W

\gcd (x^{3}, y^{2}) = \gcd (x^{2}, y^{3})

\Rightarrow \gcd ({(Π p_{i}^{a_{i}})}^{3}, {(Π p_{i}^{b_{i}})}^{2}) = \gcd ({(Π p_{i}^{a_{i}})}^{2}, {(Π p_{i}^{b_{i}})}^{3})

\Rightarrow \gcd (Π p_{i}^{3 a_{i}}, Π p_{i}^{2 b_{i}}) = \gcd (Π p_{i}^{2 a_{i}}, (Π p_{i}^{3 b_{i}})

\Rightarrow Π p_{i}^{\min (3 a_{i}, 2 b_{i})} = Π p_{i}^{\min (2 a_{i}, 3 b_{i})}

\Rightarrow \min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

P o s s i b i l i t i e s :

A : 3 a_{i} > 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} \dots (i) \\ 2 a_{i} = 3 b_{i} \dots (i i) \\ 2 a_{i} < 3 b_{i} \dots (i i i) \end{cases}

B : 3 a_{i} = 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} \dots (i) \\ 2 a_{i} = 3 b_{i} \dots (i i) \\ 2 a_{i} < 3 b_{i} \dots (i i i) \end{cases}

_{} C : 3 a_{i} < 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} \dots (i) \\ 2 a_{i} = 3 b_{i} \dots (i i) \\ 2 a_{i} < 3 b_{i} \dots (i i i) \end{cases}

A (i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} > 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 2 b_{i} = 3 b_{i} \Rightarrow b_{i} = 0

A (i i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} = 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 2 b_{i} = 3 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i} = 0

A (i i i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} < 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 2 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i}

B (i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} > 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 2 b_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i} = 0

B (i i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} = 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 2 b_{i} = 2 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i} = 0

B (i i i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} < 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 2 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i} = 0

C (i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} > 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i}

C (i i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} = 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 2 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i}

C (i i i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} < 3 b_{i}

\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})

\Rightarrow 3 a_{i} = 2 a_{i} \Rightarrow a_{i} = 0

P o s s i b l e c o n d i t i o n s f o r e x p o n e n t s o f

p r i m e f a c t o r s o f x & y

(i) a_{i} = 0 : x contain / s f a c t o r / s p_{i}^{0}

w h i l e y c o t a i n / s p_{i}^{m} w h e r e m \in W .

F o r s i m p l e e x a m p l e x = p_{i}^{0} = 1 & y = p_{i}^{m}

\gcd ({(1)}^{3}, {(p_{i}^{m})}^{2}) = \gcd ({(1)}^{2}, {(p_{i}^{m})}^{3}) = 1

(i i) b_{i} = 0 / a_{i} = b_{i} = 0 : S i m i l a r e x p l a n a t i o n

(i i i) a_{i} = b_{i} : x & y b o t h contain p_{i}^{k}

A n y c o m b i n a t i o n o f t h e s e c o n d i t i o n s

c a n b e a p p l i e d . F o r a n e x a m p l e

S o l u t i o n :

x = Π p_{i}^{0} \times Π p_{j}^{m_{j}} \times Π p_{k}^{n_{k}}

y = Π p_{i}^{l_{i}} \times Π p_{j}^{0} \times Π p_{k}^{n_{k}}

Commented by Rasheed.Sindhi last updated on 21/Jun/19

N o t e : T h e a n s w e r i s a l m o s t c o m p l e t e

a l t h o u g h i t i s n o t p e r f e c t .

T h e a n s w e r s u g g e s t a general m e t h o d h o w t o

c o m p o s e x & y t o m e e t t h e c o n d i t i o n

\gcd (x^{3}, y^{2}) = \gcd (x^{2}, y^{3})

R u l e 1 : I f a n y p r i m e f a c t o r i n x (o r y)

h a s e x p o n e n t 0, s a m e p r i m e i n y (o r i n x)

h a s a n y e x p o n e n t f r o m w h o l e n u m b e r s .

R u l e 2 : I f a n y p r i m e h a s e x p o n e n t o t h e r

t h a n 0 i n x (o r y), s a m e p r i m e i n y (o r x

h a s s a m e e x p o n e n t .

x = 3^{2} . 7^{0} . 2^{4} = 144

y = 3^{2} . 7^{3} . 2^{0} = 3087

^{∙} e x p o n e n t s e q u a l

^{∙} e x p o n e n t 0 i n a p r i m e f a c t o r o f x

A n y e x p o n e n t i n t h e same p r i m e o f y

^{∙} e x p o n e n t 0 i n a p r i m e f a c t o r o f y

A n y e x p o n e n t i n t h e same p r i m e o f x

Commented by Rasheed.Sindhi last updated on 23/Jun/19

W o r t h l e s s a n s w e r o f w o r t h l e s s q u e s t i o n!