Tuesday, 27 September 2016

elementary number theory - Find remainder when 777777 is divided by 16


Find remainder when 777777 is divided by 16.




777=48×16+9. Then 777\equiv 9 \pmod{16}.



Also by Fermat's theorem, 777^{16-1}\equiv 1 \pmod{16} i.e 777^{15}\equiv 1 \pmod{16}.




Also 777=51\times 15+4. Therefore,



777^{777}=777^{51\times 15+4}={(777^{15})}^{51}\cdot777^4\equiv 1^{15}\cdot 9^4 \pmod{16} leading to 81\cdot81 \pmod{16} \equiv 1 \pmod{16}.



But answer given for this question is 9. Please suggest.

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