Question

试说明：5的2次方 × 3的2n＋1次方 × 2的n次方,减去 3的n次方 × 6的n＋2次方能被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

上式是13的倍数,所以它能被13整除.

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