Checking divisibility of $C^{20}_7-C^{20}_8 +C^{20}_9 -C^{20}_{10}+\dots-C^{20}_{20}$ Let $N=C^{20}_7-C^{20}_8 +C^{20}_9 -C^{20}_{10}+\dots-C^{20}_{20}$. Prove that it is divisible by $3,4,7,19$.

Hint. Since by the binomial theorem 
$$0=(1-1)^{20}=\sum_{n=0}^{20}\binom{20}{n}(-1)^n=\sum_{n=0}^{6}\binom{20}{n}(-1)^n+\sum_{n=7}^{20}\binom{20}{n}(-1)^n=\sum_{n=0}^{6}\binom{20}{n}(-1)^n-N$$
it follows that 
\begin{align*}
N&=\sum_{n=0}^6\binom{20}{n}(-1)^n\\
&=1-20+190-\frac{20\cdot 19\cdot 18}{3!}+\frac{20\cdot 19\cdot 18\cdot 17}{4!}-\frac{20\cdot 19\cdot 18\cdot 17\cdot 16}{5!}+\frac{20\cdot 19\cdot 18\cdot 17\cdot 16}{6!}\\
&=-19+19\cdot 10-20\cdot 19\cdot 3+5\cdot 19\cdot 3\cdot 17- 19\cdot 3\cdot 17\cdot 16+19\cdot 17\cdot 8\cdot 15.
\end{align*}
Is it $N$ divisible by $19$ (without evaluating all these product)? 
What about $3$, $4$, and $7$?
