Question

suma a trei nr naturale mai mare ca 3 este mai mica decat 90.Aflti cele 3 nr stiind ca suma primelor 2 este de 4 ori mai mare decat al doilea numar,iar suma ultimilor 2 este de 4 ori mai mare decat primul.
VA ROG MULT

Answer 1

a+b=4b
⇒a=4b-b⇒a=3b;
b+c=4a
⇒b+c=3·4b⇒b+c=12b⇒c=12b-b⇒c=11b;
⇒a=3b b=b c=11b
⇒a+b+c=3b+b+11b=15b⇒suma celor trei numere naturale trebuie sa fie divizibila cu 15 mai mica decat 90;
⇒a+b+c∈[15,30,45,60,75]⇒5 solutii;
⇒a+b+c=15⇒15b=15⇒b=15/15⇒b=1⇒a=3b⇒a=3⇒c=11b⇒c=11;
⇒a+b+c=30⇒15b=30⇒b=30/15⇒b=2⇒a=3b⇒a=3·2⇒a=6⇒c=11b⇒ c=11 ·2⇒c=22;
⇒a+b+c=45⇒15b=45⇒b=45/15⇒b=3⇒a=3b⇒a=3·3⇒a=9⇒c=11b⇒c=11·3⇒c=33
⇒a+b+c=60⇒15b=60⇒b=60/15⇒b=4⇒a=3b⇒a=3·4⇒a=12⇒c=11b⇒c=11·4=44;
⇒a+b+c=75⇒15b=70⇒b=75/15⇒b=5⇒a=3b⇒a=3·5⇒a=15⇒c=11b⇒c=11·5=55;
⇒a+b+c∈[[3,1,11][6,2,22][9,3,33][12,4,44][15,5,55]
⇒a∈[3,6,9,12,15]⇒a∈M₃≤15;
⇒b∈[1,2,3,4,5]⇒b≤5;
⇒c∈[11,22,33,44,55]⇒c∈M₁₁≤55;