Question

Afla ultimul cifra a numarului: 2²⁰⁰⁴+3²⁰⁰⁴+5²⁰⁰⁴.
​

Answer 1

$u(2 {}^{n} ) = \begin{cases}2. \: daca \: n = 4k + 1 \\ 4. \: daca \: n = 4k + 2 \\ 8. \: daca \: n = 4 k + 3 \\ 6. \: daca \: n = 4k\end{cases} \\ \\ u(3 {}^{n} ) = \begin{cases}3. \: daca \: n = 4k + 1 \\ 9. \: daca \: n = 4k + 2 \\ 7. \: daca \: n = 4k + 3 \\ 1. \: daca \: n = 4k\end{cases} \\ \\ u(5 {}^{n} ) =\{ 5. \: daca \: n\in\mathbb{n} \\ \\ \\ \bf\color{grey}rezolvare : \\ u(u(2 {}^{2004} ) + u(3 {}^{2004} + u(5 {}^{2004} ) \\ 2004 = \color{red}4k\color{w} = 4 \times 501 \\ \iff \: = u(6 + 1 + 5) \\ \iff \: = u(12) \\ \iff \boxed{= 2}$

Answer 2

Răspuns:

2

Explicație pas cu pas:

$u(2^{1}) = 2 \\ u(2^{2}) = 4\\ u(2^{3}) = 8\\ u(2^{4}) = 6\\ u(2^{5}) = 2$

=> u(2²⁰⁰⁴) = u((2⁴)⁵⁰¹) = u(2⁴) = u(16) = 6

$u(3^{1}) = 3 \\ u(3^{2}) = 9 \\ u(3^{3}) = 7 \\ u(3^{4}) = 1 \\ u(3^{5}) = 3$

=> u(3²⁰⁰⁴) = u((3⁴)⁵⁰¹) = u(3⁴) = u(81) = 1

$u(5^{1}) = 5 \\ u(5^{2}) = 5$

=> u(5²⁰⁰⁴) = u(5) = 5

$\implies u(2²⁰⁰⁴+3²⁰⁰⁴+5²⁰⁰⁴) =$

$= u(u(2²⁰⁰⁴)+u(3²⁰⁰⁴)+u(5²⁰⁰⁴))$

$= u(6 + 1 + 5) = u(12) = \bf 2$