Question

What is the remainder of (2^91 x 6^27) divided by 7?

Answer 1

`2\equiv2\mod7`

`2^2\equiv4\mod7`

`2^3\equiv8\equiv1\mod7`


`91=3(30)+1`

`\implies2^(91)\equiv(2^3)^(30)*2^1\equiv1^(30)*2\equiv2\mod7`


`6\equiv-1\mod7`

`6^2=6*6\equiv6(-1)\equiv-6\equiv1\mod7`


`27=2(13)+1`

`\implies6^(27)\equiv(6^2)^(13)*6^1\equiv1^(13)*6\equiv6\mod7`


`\implies2^(91)*6^(27)\equiv2*6\equiv12\equiv5\mod7`