Đáp án:
Giải thích các bước giải:
Đặt A = $2^{1}$ + $2^{2}$+ $2^{3}$ +...+$2^{99}$ +$2^{100}$
⇒2A = 2($2^{1}$ + $2^{2}$+ $2^{3}$ +...+$2^{99}$ +$2^{100}$ )
⇒2A = $2^{2}$ + $2^{3}$+ $2^{4}$ +...+$2^{100}$ +$2^{101}$
⇒2A -A = ($2^{2}$ + $2^{3}$+ $2^{4}$ +...+$2^{100}$ +$2^{101}$)-($2^{1}$ + $2^{2}$+ $2^{3}$ +...+$2^{99}$ +$2^{100}$ )
⇒A = $2^{101}$ - 2
Ta có $2^{101}$ = $2^{3}$^{33} . 2$^{2}$
Mà$2^{3}$ ≡ 1(mod 7)
⇒$2^{99}$ ≡ 1(mod 7)
⇒ $2^{99}$. $2^{2}$ ≡ 4 (mod 7)
⇒ $2^{101}$ -2 ≡ 2 (mod 7)
⇒ A chia 7 dư 2
⇒$2^{1}$ + $2^{2}$+ $2^{3}$ +...+$2^{99}$ +$2^{100}$ chia 7 dư 2
