Question

Daca 2a+3b=65 si (a,b)=5, atunci determinati nr naturale a si b.

Answer 1

din (a,b)=5⇒5 I a si 5 I b⇒a=5k,b=5l,unde(k,l)=1
Atunci 2a+3b=65⇔10k+15l=65 I:5⇔2k+3l=13
din 3k<13⇒k<13/3⇒k<4+1/3,si cum k∈N⇒k∈{1,2,3,4}
1)k=1, 3l=11-imposibil,l=11/3∉N
2)k=2,3l=9⇔l=3,unde (2,3)=1
3)k=3, 3l=7-imposibil,l=7/3∉N
4)k=4, 3l=5-imposibil , l=5/3∉N
Deci (a,b)∈{(10,15)}