Mosco Olympiad (Part II)

We can observe that

$\begin{aligned} 6\times 3&=18\\ 66\times 33&=2178\\ 666\times 333&=221778\\ \cdots &=\cdots \\ \underbrace{6666666\ldots6}_{\text{n times}} \times \underbrace{3333333\ldots3}_{\text{n times}}&=\underbrace{2222222\ldots2}_{\text{(n-1) times}}1\underbrace{7777777\dots7}_{\text{(n-1) times}}8\\ \end{aligned}$

Therefore the sum of digits if $n=666$ is $665(2+7)+9=\boxed{5994}$ .

we canobserve that

$6\times 3=18\\ 66\times 33=2178\\ 666\times 333=221778\\$

if we count up the digit the pattern will go on $9\times n$ for sum of the digit Nth

so the answer would be

$9\times 666\quad =\quad 5994$