Co-prime binomial coefficient

Question

A number 'r' is prime if and only if $\binom{r-1}{k} \equiv(-1)^k \pmod r$

Since 'r' is a prime and it gives non-zero remainder by dividing $\binom{r-1}{k}$ .

So $\binom{r-1}{k}$ and 'r' are co-primes

If a₀,a₁,a₂,a₃,..........,a_r-1 are coprimes to r .

Then

Is $\binom{r-1}{0}$a₀+$\binom{r-1}{1}$a₁+$\binom{r-1}{2}$a₂+......+$\binom{r-1}{k}$a_k+......+$\binom{r-1}{r-1}$a_r-1

coprime to r ?

Answer 1

No, let $r=3,a_1=4$ and $a_0=a_2=5$ then $a_k$ is coprime to $r$, but $$ 5\binom20 +4\binom21+5\binom22=18=r^22\;. $$