Question

Chứng minh rằng 

1 +5+ 5² +5³ + ...+ 5¹⁰⁰ + 5¹⁰¹ chia hết cho 6 

Answer 1

1 +5+ 5² +5³ + ...+ 5¹⁰⁰ + 5¹⁰¹

= (1 + 5) + (5² + 5³) + ... + (5¹⁰⁰ + 5¹⁰¹)

= 6 + 5²(1 + 5) + ... + 5¹⁰⁰.(1 + 5)

= 6 + 5².6 + ... + 5¹⁰⁰.6

= 6.(1 + 5² + ... + 5¹⁰⁰) \(⋮\)6

Answer 2

\(1+5+5^2+.....+5^{101}⋮6\)

\(=\left(1+5\right)+\left(5^2+5^3\right)+.....+\left(5^{100}+5^{101}\right)\)

\(=6+\left(5^2.1+5^2.5\right)+.....+\left(5^{100}.1+5^{100}.5\right)\)

\(=6+5^2.\left(1+5\right)+.....+5^{100}.\left(1+5\right)\)

\(=6+5^2.6+....+5^{100}.6\)

\(=\left(1+5^2+....+5^{100}\right).6⋮6\)