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  1. #1
    Super Member Showcase_22's Avatar
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    mods

    Is 2008! divisible by 9^{400}?

    I figure this uses mods somewhere......can someone show me a nice place to start?
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Showcase_22 View Post
    Is 2008! divisible by 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:
    \left\lfloor \frac{2008}{3}\right\rfloor=669


    Spoiler:
    9^{400}=3^{800}
    is it possible to have 800 3's in the prime decomposition of 2008! ?
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  3. #3
    Super Member Showcase_22's Avatar
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    right, I think I getcha!

    \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]+....

    =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?
    Last edited by Showcase_22; June 1st 2009 at 12:31 PM.
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  4. #4
    Moo
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    Quote Originally Posted by Showcase_22 View Post
    right, I think I getcha!

    \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]+....

    =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
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  5. #5
    Super Member Showcase_22's Avatar
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    okay, i'll read through those threads.

    Meanwhile, I found out whose formula this is:

    De Polignac's formula - Wikipedia, the free encyclopedia
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