Question

Cho B=2^1+2^2+2^3+.....+2^30

Chứng minh rằng B chia hết cho 21

Answer 1

\(\left(2+2^3+2^5\right)+\left(2^2+2^4+2^6\right)+.........\)

\(2\left(1+2^2+2^4\right)+2^2\left(1+2^2+2^4\right)\)+...

\(2\left(21\right)+2^2\left(21\right)+....\)

21(2+2^2+...)

vậy

Answer 2

Nếu B chia hết cho 21 suy ra B chia hết cho 3,7

B=(2+2^2)+(2^3+2^4)+...+(2^29+2^30)

=2(1+2)+2^3(1+2)+...+2^29(1+2)

=2.3+2^3.3+...+2^29.3 =3(2+2^3+...+2^29) chia hết cho 3

B=(2+2^2+2^3)+...+(2^28+2^29+2^30)

=2(1+2+2^2)+...+2^28(1+2+2^2) =2.7+...+2^28.7

=7(2+...+2^28) chia hết cho 7  Vậy B chia hết cho 21