Given-a-1-3-3-2-3-3-3-4-3-100-Find-the-remainder-of-dividing-the-number-by-5-a-2-b-0-c-4-d-1-e-3-

Question Number 119956 by bobhans last updated on 28/Oct/20

G i v e n a = 1 + 3 + 3^{2} + 3^{3} + 3^{4} + \dots + 3^{100}

F i n d t h e r e m a i n d e r o f d i v i d i n g t h e n u m b e r

b y 5 .

(a) 2 (b) 0 (c) 4 (d) 1 (e) 3

Answered by talminator2856791 last updated on 08/Sep/21

a = \underset{k = 0}{\sum^{24}} 3^{4 k} (1 + 3^{2}) + \underset{k = 0}{\sum^{24}} 3^{4 k + 1} (1 + 3^{2}) + 3^{100}

\underset{k = 0}{\sum^{24}} 3^{4 k} (1 + 3^{2}) = 10 \underset{k = 0}{\sum^{24}} 3^{4 k}

\underset{k = 0}{\sum^{24}} 3^{4 k} (1 + 3^{2}) = 10 \underset{k = 0}{\sum^{24}} 3^{4 k + 1}

\to 5 ∣ 10 \underset{k = 0}{\sum^{24}} 3^{4 k}, 5 ∣ 10 \underset{k = 0}{\sum^{24}} 3^{4 k + 1}

\to 5 ∣ 1 + 3 + 3^{2} + \dots . . + 3^{99}

\Rightarrow 5 ∣ 3 (1 + 3 + 3^{2} + \dots . . + 3^{99})

= 5 ∣ 3 + 3^{2} + 3^{3} + \dots . . + 3^{100}

\Rightarrow 1 + 3 + 3^{2} + \dots \dots + 3^{100} \equiv 1 (\mod 5)

answer : d

Answered by TANMAY PANACEA last updated on 28/Oct/20

S = \frac{3^{101} - 1}{3 - 1} = \frac{1}{2} (3^{101} - 1)

3^{1} = 3

3^{2} = 9

3^{3} = 27

3^{4} = 81

3^{5} = 243

3^{6} = 729

\boldsymbol s o \boldsymbol l a s t \boldsymbol d i g i t \boldsymbol r e p e t s (3 \to 9 \to 7 \to 1)

\frac{101}{4} \to 25 c y c l e o f (3 \to 9 \to 7 \to 1) + \boldsymbol p l u s 1

\boldsymbol l a s t \boldsymbol d i g i t \boldsymbol o f 3^{101} \boldsymbol i s 3

(\boldsymbol n o w \boldsymbol l a s t \boldsymbol d i g i t 3) - 1

\boldsymbol s o \boldsymbol l a s t \boldsymbol d i g i t \boldsymbol b e c o m e s = 2

\boldsymbol w h e n \boldsymbol d e v i d e d \boldsymbol b y 2

\boldsymbol l a s t \boldsymbol d i g i t \boldsymbol b e c o m e s 1

\frac{3^{101} - 1}{2} = l a s t d i g i t = 1

\frac{3^{101} - 1}{2} = (5 \times e v e n m u l t i p l e) + \boldsymbol l a s t \boldsymbol d i g i t = 1