Question

Chứng minh rằng:

C=2 + 2² + 2³ + 2⁴ +...+2¹⁰⁰ vừa chia hết cho 31 vừa chia hết cho 5.

Answer 1

\(C=2+2^2+2^3+......+2^{100}⋮31\)

\(C=2.\left(1+2+2^2+2^3+2^4\right)+2^{95}.\left(1+2+2^2+2^3+2^4\right)\)

\(C=2.31+.......+2^{95}+31\)

\(C=31.\left(2+2^{95}\right)⋮31\)

\(\Rightarrow C⋮31\)

Answer 2

c=2^101-2 chia hết cho 31