Question

求证：5的2次方*3的2n+1次方*2的n次方-3的n次方*6的n+2次方能被13整除

Answer 1

5^2×3^(2n+1)×2^n-3^n×6^(n+2)

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

可以看出,上式是13的倍数,所以它能被13整除.