Question

ajutatima va rog din suflet va rog dau coroana​

Answer 1

$\it (a, b) = d \Rightarrow \begin{cases} \it a=dx\\ \\ \it b=dy\\ \\ (x,y)=1\end{cases}\ \ \ \ \ (1)\\ \\ \\ \big[ a,\ b ] \cdot (a,\ b)=a\cdot b\ \stackrel{(1)}{\Longrightarrow}\ \big[ a,\ b ]\cdot d = dx\cdot dy|_{:d} \Rightarrow \big[ a,\ b ]=dxy\ \ \ \ \ (2)$

$\it \big[ a,\ b ]-(a,\ b)=34\ \stackrel{(1),(2)}{\Longrightarrow}\ dxy-d=34 \Rightarrow d(xy-1)=34=2\cdot17 \Rightarrow\\ \\ \\ \Rightarrow\ \begin{cases}\it d=2\\ \\ \it xy-1=17 \Rightarrow xy=18=1\cdot18=18\cdot1=2\cdot9=9\cdot2,\(din\ (1) \end{cases}$

$\it Avem\ \ cazurile:\\ \\ I)\ d=2,\ x=1,\ y=18 \Rightarrow a=2,\ \ b=36\\ \\ II)\ d=2,\ x=18,\ y=1 \Rightarrow a=36,\ \ b=2\\ \\ III)\ d=2,\ x=2,\ y=9 \Rightarrow a=4,\ \ b=18\\ \\ IV)\ d=2,\ x=9,\ y=2 \Rightarrow a=18,\ \ b=4$

Answer 2

$\displaystyle\it\\cunoastem~relatia~:~[a.b]\cdot(a,b)=ab,~\forall a,b\in\mathbb{N}.\\notam~(a,b)=d \implies \exists m,n \in\mathbb{N},~a.i.~a=dm,~b=dn,~\underline{(m,n)=1}~.\\din~relatia:[a.b]\cdot(a,b)=ab \implies [a,b]=\frac{ab}{(a,b)}~=\frac{d^2mn}{d}=dmn.\\asadar,~[a,b]-(a,b)=dmn-d=d(mn-1)=34,~si~cum~d\in\mathbb{N},~la~fel\\si~mn-1 \implies trebuie~sa~il~scriem~pe~34~ca~produs~de~doi~factori.\\34=1\cdot34=34\cdot1=2\cdot17=17\cdot2.\\avem~doua~cazuri~de~analizat.$

$\displaystyle\it\\ cazul~1:daca~34=34\cdot1 \implies d(mn-1)=34.\\\\\boxed{daca~d=34} \implies \\mn-1=1 \implies mn=2 \implies m=2,~n=1~sau~m=1,~n=2 \implies\\\boxed{\it (a,b)\in\left\{(68,34),(34,68)\right\}}.\\\\\boxed{daca~d=1} \implies \\mn-1=34 \implies mn=35,~numarul~35~se~scrie~in~produs~de~doi~factori~\\primi~intre~ei~astfel~:~35=35\cdot1=1\cdot35=5\cdot7=7\cdot5,\\de~unde~m=35~si~n=1~sau~m=1~si~n=35~sau~m=5~si~n=7\\sau~m=7~si~n=5 \implies\\\boxed{\it (a,b)\in\left\{(35,1),(1,35),(5,7),(7,5)}~.$

$\displaystyle\it\\cazul~2:34=2\cdot17 \implies d(mn-1)=34.\\\\\boxed{\it daca~d=17} \implies\\mn-1=2 \Leftrightarrow mn=3 \implies m=3~si~n=1~sau~m=1~si~n=3 \implies\\\boxed{\it (a,b)\in\left\{(17,51),(51,17)\right\}}~.$

$\it\\\boxed{\it daca~d=2} \implies \\mn-1=17 \Leftrightarrow mn=18,~numarul~18~se~scrie~in~produs~de~doi~factori\\primi~intre~ei~astfel~:~18=1\cdot18=18\cdot1=9\cdot2=2\cdot9 \implies\\m=1~si~n=18~sau~m=18~si~n=1~sau~m=9~si~n=2~sau~\\m=2~si~n=9 \implies\\\boxed{\it(a,b)\in\left\{(2,36),(36,2),(18,4),(4,18) \right\}}~.$