Functional reminder

$f\left( x \right)$ $={ x }^{ 7 }+{ x }^{ 6 } +{ x }^{ 5 }+$ ${ x }^{ 4 }+{ x }^{ 3 }+{ x }^{ 2 }+x+1$

$\therefore f\left( { x }^{ 7 } \right) ={ x }^{ 13 }+{ x }^{ 12 }+{ x }^{ 11 }+{ x }^{ 10 }+{ x }^{ 9 }+{ x }^{ 8 }+1$

$={ x }^{ 13 }+{ x }^{ 12 }+{ x }^{ 11 }+{ x }^{ 10 }+{ x }^{ 9 }+{ x }^{ 8 }+{ x }^{ 7 }+1-{ x }^{ 7 }$

$={ x }^{ 7 }({ x }^{ 6 }+{ x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 3 }+{ x }^{ 2 }+x+1)+1-{ x }^{ 7 }$

$={ x }^{ 7 }\times f\left( x \right) +1-{ x }^{ 7 }$

Now by dividing with $f\left( x \right)$ we get the reminder $1-{ x }^{ 7 }$