Find out x,y, such that gcd(x^3 ,y^2 )=gcd(x^2 ,y^3 )


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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
	$x=Πp_i ^a_i y=Πp_i ^b_i : p_i ∈P ∧ a_i ,b_i ∈W gcd(x^3 ,y^2 )=gcd(x^2 ,y^3 ) ⇒gcd( (Πp_i ^a_i )^3 ,(Πp_i ^b_i )^2 )=gcd( (Πp_i ^a_i )^2 ,(Πp_i ^b_i )^3 ) ⇒gcd( Πp_i ^(3a_i ) , Πp_i ^(2b_i ) )=gcd( Πp_i ^(2a_i ) , (Πp_i ^(3b_i ) ) ⇒Πp_i ^(min(3a_i , 2b_i )) =Πp_i ^(min(2a_i , 3b_i )) ⇒min(3a_i , 2b_i )=min(2a_i , 3b_i ) Possibilities: A: 3a_i >2b_i ∧ { ((2a_i >3b_i ...(i))),((2a_i =3b_i ...(ii))),((2a_i <3b_i ...(iii))) :} B: 3a_i =2b_i ∧ { ((2a_i >3b_i ...(i))),((2a_i =3b_i ...(ii))),((2a_i <3b_i ...(iii))) :} _ C: 3a_i <2b_i ∧ { ((2a_i >3b_i ...(i))),((2a_i =3b_i ...(ii))),((2a_i <3b_i ...(iii))) :} A(i): 3a_i >2b_i ∧ 2a_i >3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒2b_i =3b_i ⇒b_i =0 A(ii): 3a_i >2b_i ∧ 2a_i =3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒2b_i =3b_i =2a_i ⇒a_i =b_i =0 A(iii): 3a_i >2b_i ∧ 2a_i <3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒2b_i =2a_i ⇒a_i =b_i B(i): 3a_i =2b_i ∧ 2a_i >3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =2b_i =3b_i ⇒a_i =b_i =0 B(ii): 3a_i =2b_i ∧ 2a_i =3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =2b_i =2a_i =3b_i ⇒a_i =b_i =0 B(iii): 3a_i =2b_i ∧ 2a_i <3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =2b_i =2a_i ⇒a_i =b_i =0 C(i): 3a_i <2b_i ∧ 2a_i >3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =3b_i ⇒a_i =b_i C(ii): 3a_i <2b_i ∧ 2a_i =3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =2a_i =3b_i ⇒a_i =b_i C(iii): 3a_i <2b_i ∧ 2a_i <3b_i min(3a_i , 2b_i )=min(2a_i , 3b_i ) ⇒3a_i =2a_i ⇒a_i =0 Possible conditions for exponents of prime factors of x & y (i) a_i =0 : x contain/s factor/s p_i ^0 while y cotain/s p_i ^m where m∈W. For simple example x= p_i ^0 =1& y=p_i ^m gcd((1)^3 ,(p_i ^m )^2 )=gcd((1)^2 ,(p_i ^m )^3 )=1 (ii) b_i =0/a_i =b_i =0 : Similar explanation (iii) a_i =b_i :x & y both contain p_i ^k Any combination of these conditions can be applied. For an example Solution: x=Πp_i ^0 ×Πp_j ^m_j ×Πp_k ^n_k y=Πp_i ^l_i ×Πp_j ^0 ×Πp_k ^n_k$
	$x = Π p_{i}^{a_{i}} y = Π p_{i}^{b_{i}} : p_{i} \in P \land a_{i}, b_{i} \in W$ $\gcd (x^{3}, y^{2}) = \gcd (x^{2}, y^{3})$ $\Rightarrow \gcd ({(Π p_{i}^{a_{i}})}^{3}, {(Π p_{i}^{b_{i}})}^{2}) = \gcd ({(Π p_{i}^{a_{i}})}^{2}, {(Π p_{i}^{b_{i}})}^{3})$ $\Rightarrow \gcd (Π p_{i}^{3 a_{i}}, Π p_{i}^{2 b_{i}}) = \gcd (Π p_{i}^{2 a_{i}}, (Π p_{i}^{3 b_{i}})$ $\Rightarrow Π p_{i}^{\min (3 a_{i}, 2 b_{i})} = Π p_{i}^{\min (2 a_{i}, 3 b_{i})}$ $\Rightarrow \min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $P o s s i b i l i t i e s :$ $A : 3 a_{i} > 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} . . . (i) \\ 2 a_{i} = 3 b_{i} . . . (i i) \\ 2 a_{i} < 3 b_{i} . . . (i i i) \end{cases}$ $B : 3 a_{i} = 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} . . . (i) \\ 2 a_{i} = 3 b_{i} . . . (i i) \\ 2 a_{i} < 3 b_{i} . . . (i i i) \end{cases}$ $_{} C : 3 a_{i} < 2 b_{i} \land {\begin{cases} 2 a_{i} > 3 b_{i} . . . (i) \\ 2 a_{i} = 3 b_{i} . . . (i i) \\ 2 a_{i} < 3 b_{i} . . . (i i i) \end{cases}$ $A (i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} > 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 2 b_{i} = 3 b_{i} \Rightarrow b_{i} = 0$ $A (i i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} = 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 2 b_{i} = 3 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i} = 0$ $A (i i i) : 3 a_{i} > 2 b_{i} \land 2 a_{i} < 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 2 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i}$ $B (i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} > 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 2 b_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i} = 0$ $B (i i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} = 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 2 b_{i} = 2 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i} = 0$ $B (i i i) : 3 a_{i} = 2 b_{i} \land 2 a_{i} < 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 2 b_{i} = 2 a_{i} \Rightarrow a_{i} = b_{i} = 0$ $C (i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} > 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i}$ $C (i i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} = 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 2 a_{i} = 3 b_{i} \Rightarrow a_{i} = b_{i}$ $C (i i i) : 3 a_{i} < 2 b_{i} \land 2 a_{i} < 3 b_{i}$ $\min (3 a_{i}, 2 b_{i}) = \min (2 a_{i}, 3 b_{i})$ $\Rightarrow 3 a_{i} = 2 a_{i} \Rightarrow a_{i} = 0$ $P o s s i b l e c o n d i t i o n s f o r e x p o n e n t s o f$ $p r i m e f a c t o r s o f x & y$ $(i) a_{i} = 0 : x contain / s f a c t o r / s p_{i}^{0}$ $w h i l e y c o t a i n / s p_{i}^{m} w h e r e m \in W .$ $F o r s i m p l e e x a m p l e x = p_{i}^{0} = 1 & y = p_{i}^{m}$ $\gcd ({(1)}^{3}, {(p_{i}^{m})}^{2}) = \gcd ({(1)}^{2}, {(p_{i}^{m})}^{3}) = 1$ $(i i) b_{i} = 0 / a_{i} = b_{i} = 0 : S i m i l a r e x p l a n a t i o n$ $(i i i) a_{i} = b_{i} : x & y b o t h contain p_{i}^{k}$ $A n y c o m b i n a t i o n o f t h e s e c o n d i t i o n s$ $c a n b e a p p l i e d . F o r a n e x a m p l e$ $S o l u t i o n :$ $x = Π p_{i}^{0} \times Π p_{j}^{m_{j}} \times Π p_{k}^{n_{k}}$ $y = Π p_{i}^{l_{i}} \times Π p_{j}^{0} \times Π p_{k}^{n_{k}}$
	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

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