Question

[Stella]

Cho A = 2 + 2² + 2³ + ... + 2¹⁰⁰

CMR : A chia hết cho 7 ; 17 ; 31

Answer 1

[Lucy Heartfilia]

A = 2 + 2² + 2³ + ... + 2¹⁰⁰

A = ( 2 + 2² + 2³ ) + ... + ( 2⁹⁸ + 2⁹⁹ + 2¹⁰⁰ )

A = 2 . ( 1 + 2 + 4 ) + .... + 2⁹⁸ . ( 1 + 2 + 4 )

A = 2 . 7 + ... + 2⁹⁸ . 7

A = ( 2 + ... + 2⁹⁸ ) . 7 chia hết cho 7 .

A = 2 + 2² + 2³ + ... + 2¹⁰⁰ 

A = ( 2 + 2² + 2³ + 2⁴ ) + ... + ( 2⁹⁷ + 2⁹⁸ + 2⁹⁹ + 2¹⁰⁰ )

A = 2 . ( 1 + 2 + 4 + 9 ) + ... + 2⁹⁷ . ( 1 + 2 + 4 + 9 )

A = 2 . 15 + ... + 2⁹⁷ . 15

A = ( 2 + ... + 2⁹⁷ ) . 15 chia hết cho 15

A = 2 + 2² + 2³ + ... + 2¹⁰⁰

A = ( 2 + 2² + 2³ + 2⁴ + 2⁵ ) + ... + ( 2⁹⁶ + 2⁹⁷ + 2⁹⁸ + 2⁹⁹ + 2¹⁰⁰ )

A = 2 . ( 1 + 2 + 4 + 9 + 16 ) + ... + 2⁹⁶ . ( 1 + 2 + 4 + 9 + 16 )

A = 2 . 31 + ... + 2⁹⁶ . 31

A = ( 2 + ... + 2⁹⁶ ) . 31 chia hết cho 31