1. ## mods

Is 2008! divisible by $\displaystyle 9^{400}$?

I figure this uses mods somewhere......can someone show me a nice place to start?

2. Hello,
Originally Posted by Showcase_22
Is 2008! divisible by $\displaystyle 9^{400}$?

I figure this uses mods somewhere......can someone show me a nice place to start?
How many multiples of 3 are there between 1 and 2009 ?
Spoiler:
$\displaystyle \left\lfloor \frac{2008}{3}\right\rfloor=669$

Spoiler:
$\displaystyle 9^{400}=3^{800}$
is it possible to have 800 3's in the prime decomposition of 2008! ?

3. right, I think I getcha!

$\displaystyle \left[\frac{2008}{3} \right]+\left[ \frac{2008}{3^2} \right]+\left[ \frac{2008}{3^3}\right]+\left[ \frac{2008}{3^4} \right]+\left[ \frac{2008}{3^5} \right]+\left[ \frac{2008}{3^6} \right]+\left[ \frac{2008}{3^7} \right]+\left[ \frac{2008}{3^8} \right]+....$

$\displaystyle =669+223+74+24+8+2+0+....>800$

(Whose formula is this known as? Is it De pognac's?)

So it is possible to have 800 3's in the prime decomposition of 2008!

Thanks Moo

P.S: How do you get those really cool spoiler windows to appear?

4. Originally Posted by Showcase_22
right, I think I getcha!

$\displaystyle \left[\frac{2008}{3} \right]+\left[ \frac{2008}{3^2} \right]+\left[ \frac{2008}{3^3}\right]+\left[ \frac{2008}{3^4} \right]+\left[ \frac{2008}{3^5} \right]+\left[ \frac{2008}{3^6} \right]+\left[ \frac{2008}{3^7} \right]+\left[ \frac{2008}{3^8} \right]+....$

$\displaystyle =669+223+74+24+8+2+0+....>800$
Yes

Whew...I struggled with finding these threads :
http://www.mathhelpforum.com/math-he...orization.html
http://www.mathhelpforum.com/math-he...ation-2-a.html
They may give you further insight on the formula

(Whose formula is this known as? Is it De pognac's?)
I don't know the name
But I looked for de Pognac and didn't find anything significant

P.S: How do you get those really cool spoiler windows to appear?
With the [spoiler][/spoiler] tags