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Prime divisors of factorials

  1. Oct 17, 2004 #1
    I was reading this tutorial and I came across this part which I didn't quite understand:

    I don't follow this. What does "a will be counted j times and will contribute i towards t" mean? Why does this show that t=s?
     
  2. jcsd
  3. Oct 17, 2004 #2
    First get this clear ...
    m1 denotes the number of values in {1,..,n} divisible by p
    m2 denotes the number of values in {1,..,n} divisible by p^2
    m3 denotes the number of values in {1,..,n} divisible by p^3
    ...
    ...
    mk denotes the number of values in {1,..,n} divisible by p^k

    if a is divisible by p^j and not by p^j+1
    that means the maximum power of p that can divide a is j ...

    if p^j|a then p|a,p^2|a,p^3|a,p^4|a,.....,p^j|a

    that means this number a will have been counted in m1,m2,m3,m4,....,mj
    t = sum of all m_i's
    so a will have been counted j times in t ....

    or in other words contribute an m_i towards t
    (can u see why?)

    set a=n and apply floor

    -- AI
     
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