# Thread: equatle square root of huge number

1. ## equatle square root of huge number

2. Originally Posted by hotgal24
*Ahem* 333...33 (1000 digits)

3. Note that $\displaystyle \underbrace{111\ldots111}_{\textrm{$n$digits}}\ =\ \frac{10^n-1}{9}$.

Hence

$\displaystyle \underbrace{111\ldots111}_{\textrm{2$n$digits}}-\underbrace{222\ldots222}_{\textrm{$n$digits}}\ =\ \frac{10^{2n}-1}{9}\,-\,2\cdot\frac{10^n-1}{9}$

$\displaystyle \color{white}.\hspace{42mm}.$ $\displaystyle =\ \frac{\left(10^n+1\right)\left(10^n-1\right)}{9}-\frac{2\left(10^n-1\right)}{9}$

$\displaystyle \color{white}.\hspace{42mm}.$ $\displaystyle =\ \frac{\left(10^n-1\right)\left(10^n+1-2\right)}{9}$

$\displaystyle \color{white}.\hspace{42mm}.$ $\displaystyle =\ \left(\frac{10^n-1}{3}\right)^2$

Substitute $\displaystyle n=1000$ and you have your answer.

4. Originally Posted by mr fantastic
*Ahem* 333...33 (1000 digits)
Some hints:

222...22 = 2(111...11) (1000 digits)

111...11 (2000 digits) divided by 111...11 (1000 digits) = 1000...001 (1001 digits)

1000...001 - 2 = 999...99 = 9(111...11) (1000 digits)