For integers a and b; and for integer m > 0 we write a\equivb(\mod m) if m divides a-b.if 8^2017\equivn(\mod13) and 0 \leq n

Correct Answer is : none of the above

AP ICET 2017 Shift 2 Paper

Question : 109 of 200

Marks: +1, -0

For integers a and b; and for integer

m > 0

we write

a≡b(\mod\;m)

if m divides a-b.if

8^2017≡n(\mod13)

and

0 ≤ n < 13

then

n=

Solution:

The cyclicity of 8p is 4 i.e., the remainder repeats in cycle of 4 when divided by 13.
2017

=(504×4)+1

Remainder

8^1=8\;\;\;\;\;\;\;\;\;\;\;\;8

8^{2}=64\;\;\;\;\;\;\;\;\;12

8^{3}=512\;\;\;\;\;\;\;\;\;5

8^{4}=\4096\;\;\;\;\;\;\;1

8^{5}=32768\;\;\;\;\;8

8^{6}=262144\;\;\;12

∴8^{2017}=8^{(504*4)+1}

Remainder of 82017 is same as remainder of 81 = 8
Hence

n=8.

Aliter :

(8)^{2017}&=8×8^{2016}=8(64)^{1008}

R[8×(64)^{1008}]=8×(-1)^{1008}=8

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