Question

suma a doua numere este 47 iar suma rasturnatelor lor este 173 determinati nr

Answer 1

[tex]Fie\ \overline{ab}\ si\ \overline{cd}\ numerele:\\ \overline{ab}+\overline{cd}=47\\ \overline{ba}+\overline{dc}=173\\ ..........................\\ 10a+b+10c+d=47\\ 10b+a+10d+c=173\\ ................................\\ Daca\ le\ adunam:\\ 11a+11b+11c+11d=220\\ 11(a+b+c+d)=220\\ a+b+c+d=20\\ Daca\ le\ scadem:\\ 9b-9a+9d-9c=126\\ 9(b-a+d-c)=126\\ b-a+d-c=14\\ Adunam\ cele\ doua\ relatii:\\ 2d+2b=34\\ b+d=17\\ Cum\ a,b\leq 9,\ deducem\ doua\ solutii\ posibile:\\ a=9,b=8\ si\ a=8,b=9\\ [/tex]
[tex]Inlocuim:\\ 17+a+c=20\\ a+c=3\\ Cum:a,c\geq 1\, deducem\ doua\ cazuri\ posibile:\\ a=1,c=2\ si\ a=2,c=1.\\ Asadar\ solutiile\ ecuatiei\ sunt:\\ S:(\overline{ab},\overline{cd})\in \{(18,29) ;( 19,28); (28, 19) ;( 29, 18)\}[/tex]