Đáp án+ Giải thích các bước giải:

a) A = 1 + 2 + 2² + 2³ + ... + $2^{2019}$ 

A = (1 + 2) + (2² + 2³) + ... ($2^{2018}$ + $2^{2019}$)

A = 1.(1 + 2) + 2².(1 + 2) + ... + $2^{2018}$ . (1 + 2)

A = 1 · 3 + 2² · 3 + .... + $2^{2018}$ · 3

A = 3 · (1 + 2² + ... + $2^{2018}$)

Vậy A chia hết cho 3.

b) A = 1 + 2 + 2² + 2³ + ... + $2^{2019}$ 

A = (1 + 2 + 2² + 2³) + ... +($2^{2016}$ + $2^{2017}$ + $2^{2018}$ + $2^{2019}$)

A = 1 · (1 + 2 + 2² + 2³) + ... + $2^{2016}$ · (1 + 2 + 2² + 2³)

A = 1 · 15 + .... + $2^{2016}$ · 15

A = 15 · (1 + ... + $2^{2016}$)

A = 3 · 5 · (1 + ... + $2^{2016}$)

Vậy A chia hết cho 5.

c) A = 1 + 2 + 2² + 2³ + ... + $2^{2019}$ 

A = (1 + 2 + 2² + 2³ + $2^{4}$ )+ ... +($2^{2015}$ $2^{2016}$ + $2^{2017}$ + $2^{2018}$ + $2^{2019}$)

A = 1 · (1 + 2 + 2² + 2³ + $2^{4}$ ) + ... + $2^{2015}$ · (1 + 2 + 2² + 2³ + $2^{4}$)

A = 1 · 31 + .... + $2^{2015}$ · 31

A = 31 · (1 + ... + $2^{2015}$)

Vậy A chia hết cho 31.