Question

Chứng minh:

a) 2⁰ + 2¹ + 2² + ... + 2^{5n - 3} + 2^{5n - 3} + 5^{5n -1}  chia hết cho 31

Answer 1

Đặt A = 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + ..... +2^{5n-6 +} 2^5n-5 + 2^5n-4 + 2^5n-3 + 2^5n-2 + 2^5n-1

=> A = ( 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ ) + ..... + ( 2^5n-6 + 2^5n-5 + 2^5n-4 + 2^5n-3 + 2^5n-2 + 2^{5n-1 )}

=> A = 2⁰ ( 1 + 2¹ + 2² + 2³ + 2⁴ ) + ..... + 2^5n-6 ( 1 + 2¹ + 2² + 2³ + 2⁴ )

=> A = 1.31 + 2⁵ .31 + ..... + 2^5n-6.31 

=> A = 31.( 1 + 2⁵ + ..... + 2^5n-6 )

Vì 31 ⋮ 31 => A ⋮ 31 ( đpcm )