Question

求证N=5^2*2^2n+1*2^n-3^n*3^n*6^n+1能被13整除

Answer 1

证明：

5^2×3^(2n+1)×2^n-3^n×6^(n+2)

=5^2×3^(2n+1)×2^n-3^n×(2×3)^(n+2)

=5^2×3^(2n+1)×2^n-3^n×2^(n+2)×3^(n+2)

=5^2×3^(2n+1)×2^n-3^(2n+2)×2^(n+2)

=5^2×3^(2n+1)×2^n-3^(2n+1)×3×2^n×2^2

=3^(2n+1)×2^n×[5^2-3×2^2]

=3^(2n+1)×2^n×[25-12]

=3^(2n+1)×2^n×13