Question

determina x si y care verifica egalitatea x×y+cmmmcx,y=90

Answer 1

xy+[x,y]=90
[x.y]+(x.y)[x,y]=90
[x,y]((x.y)+1)=90
deci
[x,y]|90
[x,y]∈D90 N={1;2;3;5;6;9;10;15;18;30;45;90}
[x.y]=1⇒x=y=1nu aresens
[x;y]=2⇒(x,y)=90:2-1=44>2 ,(x,y)>[x,y] contradictie
[x,y]=3⇒(x,y)=90:3-1=29>3 contradictie
...................................
[x,y]=6⇒(x,y)=90:6-1=14>6

[x,y]=9⇒(x,y)=90:9-1=9⇒x=y=9 care verifica 9*9+9=90

[x,y]=10⇒(x,y)=90:10-1=8⇒nu avem x,y, pt ca (x,y) contine pe 2³si [x;y] contine[x;y]

[x,y]=15⇒(x,y)=90:15-1=5⇒x=5 si y=15 (saux=15 siy=5) care verifica
                                                              5*15+15=75+15=90

[x,y]=18⇒(x,y)=90:18-1=5-1=4⇒nu exista x si y pt ca 4=2² si 18=2*3²
[x,y]=30⇒(x.y)=90:30-1=2⇒x=2 si y=30 sau x=30 si y=2
                                             verificare 2*30+30=90


[x,y]=45 ⇒(x,y)=90:45-1=2-1=1⇒x=1 y=45 sau x=45 ,y=1
                                                     x=5, y=9 sau x=9, y=5

Deci (x,y)∈{(9;9);(5;15);(15;5);(2;30);(30;2);(1;45);(45;1);(5;9);(9;5)}