Question

Sa se afle suma numerelor care impartite la 9 dau catul 12

Answer 1

$D = I\cdot C+R \\ \\ D = 9\cdot 12+R,\quad 0\leq R\ \textless \ 9\\ \\ \Rightarrow S = \Big(9\cdot 12+0\Big)+\Big(9\cdot 12+1\Big)+\Big(9\cdot 12+2\Big)+...+\Big(9\cdot 12+8\Big)\Rightarrow \\ \\ S = 9\cdot 12+9\cdot 12+\underset{de~ 9~ ori}{\underbrace{...}}+9\cdot 12 +0+1+2+...+8 \\ \\ S = 9\cdot 12\cdot 9+1+2+...+8 \\ \\ S = 9\cdot 12\cdot 9+\dfrac{8\cdot (8+1)}{2} \\ \\ S = 972+\dfrac{8\cdot 9}{2} \\ \\ S = 972+4\cdot 9\\ \\ S = 972+36 \\ \\ \boxed{S = 1008}$