Question

35. Fie proporţia
2/a=3^50 +3^49+3^48+…+3^2+3 totul pe b Aflați ultima cifră a numărului a*b

Answer 1

Răspuns:

$U(a\cdot b)=4$

Explicație pas cu pas:

$3+3^2+...+3^{50}=S\\3^2+3^3+...+3^{51}=3S\\2S=3^{51}+...+3^3+3^2-(3^{50}+...+3^2+3)=3^{51}-3\\\frac{a}{2}=\frac{S}{b} \implies a\cdot b=2\cdot S$

$3^1=3\\3^2=9\\3^3=27\\3^4=81\\...\\U(3^n)=\left\{\begin{array}{lr} 1, & n=4k\\ 3, & n=4k+1\\ 9, & n=4k+2\\ 7, & n=4k+3 \end{array}$

$U(3^{51}-3)=U(3^{12\cdot4+3}-3)=7-3=4 \\\boxed{U(a\cdot b)=4}$