Find the remainder when 2256 is divided by 17.
A. 1
B. 16
C. 14
D. None of these
Answer: Option A
Solution(By Examveda Team)
$$\eqalign{ & {\text{Given}}, \cr & \frac{{{2^{256}}}}{{17}} \cr & {\text{We}}\,{\text{can}}\,{\text{write}}\,{\text{it}}\,{\text{as}}:\,{\left( {{2^4}} \right)^{64}} \cr & \frac{{{{16}^{64}}}}{{17}} \cr & {\text{Individually, when 16 is divided by 17,}} \cr & {\text{gives a negative reminder of - 1}}{\text{.}} \cr & {\text{Required Remainder}}, \cr & {\left( { - 1} \right)^{64}} = 1 \cr & {\text{Alternatively}}, \cr & \frac{{{{16}^{64}}}}{{17}},\,{\text{can}}\,{\text{be}}\,{\text{written}}\,{\text{as}} \cr & \frac{{\left( {16 \times 16 \times 16 \times 16 \times 16\,......\,64\,{\text{times}}} \right)}}{{17}} \cr & {\text{Now,}}\,{\text{we}}\,{\text{take}}\,{\text{the}}\,{\text{negative}}\,{\text{remainder}}\,{\text{of}}\,{\text{each,}} \cr & {\text{16}}\,{\text{divided}}\,{\text{by}}\,{\text{17}}\,{\text{gives}}\,{\text{negative}}\,{\text{remainder}}\,{\text{ - 1}}{\text{.}} \cr & {\text{So,}}\,{\text{the}}\,{\text{Remainder}}\,{\text{will}}\,{\text{be}} \cr & \left( { - 1 \times - 1 \times - 1 \times - 1 \times - 1\,.......\,64\,{\text{times}}} \right) \cr & = 1 \cr} $$Join The Discussion
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Related Questions on Number System
Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:
A. 12, 24, 36
B. 11, 22, 33
C. 12, 24, 32
D. 5, 10, 15
2(4x64)/17
2x2x2x2=16/17
16/17 = -1
:. -1+17= 16
The correct answer is 16
16