Question

证明2的10次方-2的8次方+2的6次方-2的4次方+2的2次方-1能被9整除

Answer 1

原式=2^8(2^2-1)+2^4(2^2-1)+2^2-1

=(2^2-1)(2^8+2^4+1)

=3[2^8+2*2^4+1-2^4]

=3[(2^4+1)^2-2^4]

=3[(2^4+2^2+1)(2^4-2^2+1)]

=3[(2^4+2*2^2+1-2^2)](2^4-2^2+1)

=3[(2^2+1)^2-2^2](2^4-2^2+1)

=3(2^2+2+1)(2^2-2+1)(2^4-2^2+1)

=9(2^2+2+1)(2^4-2^2+1)

∴命题成立