3.1.15.
aabb=1000a+100a+10b+b=1100a+11b=11(100a+b)
a)N=aabb=11(100a+b), deci N este divizibil cu 11
b)a+b=11 =>11(100a+b)=11(99a+a+b)=11(11•9•a +11)=11[11(9a+1)]=121(9a+1), deci este divizibil cu 121 daca a+b=11
c)N patrat perfect => 11(100a+b)=patrat perfect => 100a+b=11•x, x= patrat perfect de 2 cifre
x∈ {16; 25; 36; 49; 64; 81}
convine doar 11•64=704 => 11(100a+b)=11•11•64; este patrat perfect
100a+b=704; a=7 si b=4
=> aabb=7744; 7744=88²
3.1.20
N=axyb+byxa
N=1000a+100x+10y+b+1000b+100y+10x+a
N=1001a+1001b+110x+110y
N=1001(a+b) +110(x+y)
1001=7•11•13
N divizibil cu 13, 1001(a+b) divizibil cu 13 => 110(x+y) divizibil cu 13
(110, 13)=1
=> x+y=13
(x, y)∈ {(4; 9}, (5; 8), (6; 7), (7; 6}, (8; 5), (9; 4)}
