Question

Calculati suma tuturor numerelor abab(in baza zece), stiind ca ab-ba=a+3b

Answer 1

ab-ba=a+3b ⇔
10a+b-(10b+a)=a+3b ⇔
9a-9b=a+3b ⇔
9(a-b)=a+3b ⇔
9a-9b-a-3b=0 ⇔
8a-12b=0 ⇔
4(2a-3b)=0 ⇒2a-3b=0 ⇒2a=3b unde a;b∈N si a;b=cifre nenule;
Avem urmatoarele solutii:
a=3 ⇒b=2;
a=6 ⇒b=4;
a=9 ⇒b=6;
Numerele de forma abab cu bara deasupra care respecta cerinta de mai sus sunt
abab∈{3232;6464;9696}
S=3232+6464+9696
S=19392 ;