Question

. What is the remainder when 4^1000 is
divisible by 7?
00 [CDS 2014]
(a) 1
(6) 2
(c) 4
(d) None of these​

Answer 1

Answer:

option. c

Step-by-step explanation:

1000/4= no reminder

4^4=256

remaining 4

Answer 2

Answer:

4(c)

${4}^{1000} = {2}^{2000} \\ = {2}^{2} \times {2}^{1998} \\ = 4( {2}^{3} ) {}^{666} \\ =4 (7 + 1) {}^{666} \\ use \: \: binomial \\ \\ (7 + 1) {}^{666} = \sum _{k = 0}^{666 } \binom{666}{k} {7}^{k} = 1 + 7k$

So your number is some

$= 4(1 + 7k) \: \: \: for \: \: some \: \: k \in \: z \\ = 4 + 28k$

So remainder will be 4 .