Cocassi Function Algebra New Year Special

Algebra Level 3

Let $F(x)$ denote a monic $7^\text{th}$ degree polynomial with satisfying the following

$F(2) = 2, F(3) = 3, F(4) =4, F(5) = 5, F(6) = 6, F(7) =7, F(8) = 8$

What is the value of $F(9) \bmod 7$ ?

1 solution

Chew-Seong Cheong
Jan 6, 2015

Since $F(x)$ is a monic seventh degree polynomial. It can be written as:

$F(x) = (x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)+x$

Therefore,

$F(9) = (9-2)(9-3)(9-4)(9-5)(9-6)(9-7)(9-8)+9$

$= 7!+9 = 5049$

Now to solve for:

$5049 \pmod {7} \equiv (7!+9) \pmod{7} \equiv 9 \pmod{7} \equiv \boxed {2} \pmod {7}$