evaluating remainders by using binomial

Question

I am finding the remainder when $6^{88}$ is divided by $19$ , I applied Little theorem and obtained $9$, which is correct. My question is, can this be solved by using binomial theorem? I am not getting how to approach this.

Answer 1

\begin{eqnarray} 6^{88} &=& 36^{44} \\ &=& (38-2)^{44}\\ &=& 38^{44}-44\cdot 38^{43}\cdot 2 + ...-44\cdot 38 \cdot 2^{43} +2^{44}\\ &=& 19a+2^{44} \end{eqnarray}

So you reduce your problem to $2^{44} = 16^{11} = (19-3)^{11} = 19b - 3^{11}$...