Interesting number theory pattern with proof

Question

I have found the following number pattern to hold true $$333*576=191808$$ $$333000333*576=191808191808$$ $$333000333000333*576=191808191808191808$$… And so on. My question, then, is how might I prove that this pattern holds, rather than simply stating it does. Any help is appreciated!

Answer 1

You can write the products alternatively as \begin{eqnarray*} 333\times576&=&191808\times1&=&191808\\ 333000333\times576&=&191808\times1000001&=&191808191808\\ 333000333000333\times576&=&191808\times1000001000001&=&191808191808191808 \end{eqnarray*} This is independent of the six-digit number $191808$ or its particular factorization $333\times576$. Is there anything more to prove here?