Question

cho A=2+2²+2³+........+2⁶⁰.Chung minh rang A chia het cho 5.31 va 15

Cho B =1+2+2²+2³+.........+2⁶⁰.Tim so du khi chia B cho 5,31 va 15

Answer 1

A = \(\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57} +2^{58}+2^{59}+2^{60}\right)\)

\(=2.\left(1+2+2^2+2^3\right)+2^5.\left(1+2+2^2+2^3\right)+..2^{57}.\left(1+2+2^2+2^3\right)\)

\(=2.15+2^5.15+...+2^{57}.15\)

\(=15.\left(2+2^5+...+2^{57}\right)\text{chia hết cho 15}\)

\(=5.3.\left(2+2^5+...+2^{57}\right)\text{ chia hết cho 5}\left(1\right)\)

A = \(2.\left(1+2+2^2+2^3+2^4\right)+2^6.\left(1+2+2^2+2^3+2^4\right)+...+2^{56}.\left(1+2+2^2+2^3+2^4\right)\)

\(=2.31+2^6.31+...+2^{56}.31\)

\(=31.\left(2+2^6+...+2^{56}\right)\text{ chia hết cho 31}\left(2\right)\)

Từ (1) và (2) => A chia hết cho 5.31

B = 1 + A nên B chia 5,31 và 15 đều dư 1.

 

Answer 2

\(\frac{7}{58}\)

Answer 3

\(\frac{7}{58}\)