A number theory problem by Gajender Singh
Number Theory
Level
2
Find the remainder when 2^86 is divided by 7.
The answer is 4.
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1 solution
\frac { { 2 }^{ 86 } }{ 7 } =\frac { { 2 }^{ 84 }{ 2 }^{ 2 } }{ { 2 }^{ 3 }-1 } =\frac { { 2 }^{ { 3 }^{ 28 } }.4 }{ { 2 }^{ 3 }-1 } \qquad \ By\quad remainder\quad theorem,\quad if\quad x={ 2 }^{ 3 }\quad and\quad f(x)=4{ x }^{ 28 }\ Required\quad remainder=f(1)=4{ (1) }^{ 28 }=4.