An inequality in numbers Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 digits}$? Why? How much is it larger?
 A: note that $$\underbrace{888\cdots8}_\text{19 digits}=2*\underbrace{444\cdots4}_\text{19 digits}$$
and $$2*\underbrace{333\cdots3}_\text{68 digits}=\underbrace{666\cdots6}_\text{68 digits}$$
further
$$\underbrace{666\cdots6}_\text{68 digits}+1=\underbrace{666\cdots7}_\text{68 digits}$$
Thus
$$\underbrace{888\cdots8}_\text{19 digits}*\underbrace{333\cdots3}_\text{68 digits}=\underbrace{444\cdots4}_\text{19 digits}*2*\underbrace{333\cdots3}_\text{68 digits}=
\underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots6}_\text{68 digits}$$
which is by $\underbrace{444\cdots4}_\text{19 digits}$ smaller than
$$\underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots7}_\text{68 digits}=\underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots6}_\text{68 digits}+$\underbrace{444\cdots4}_\text{19 digits}$$
A: Let those four numbers be $a,b,c,d$ respectively. Then $a=2c$ and $d=2b+1$. So $cd-ab=c$.
A: You have
$\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits} = 
8\cdot 3 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits}) = 
24 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits})$.
Similarly we get
$\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots6}_\text{68 digits} =
4\cdot 6 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits}) =
24 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits})$.
So these two numbers are equal.
It is clear that 
$$\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots6}_\text{68 digits} \le \underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots7}_\text{68 digits}.$$
Since the multiplier is increased by one, the difference is exactly $\underbrace{444\cdots4}_\text{19 digits}$.
