Question

 

4. Hãy chứng tỏ rằng:

a, A=7¹⁵+7¹⁶+7¹⁷ chia hết cho 57

b, B=2+2²+2³+2⁴+...2⁶⁰chia hết cho 7

Answer 1

\(a,A=7^{15}+7^{16}+7^{17}\)

\(A=7^{15}\left(1+7+7^2\right)\)

\(A=7^{15}.57\)

Ta có :

\(A=7^{15}.57⋮57\)

\(\Rightarrow A⋮57\)

Answer 2

\(b,B=2+2^2+2^3+....+2^{60}\)

\(B=\left(2+2^2+2^3\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(B=2\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)

\(B=2.7+...+2^{58}.7\)

\(B=7\left(2+2^4+....+2^{58}\right)\)

Ta có :

\(B=7\left(2+2^4+....+2^{58}\right)⋮7\)

\(\Rightarrow B⋮7\)