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#1 2007-02-02 09:18:05

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Check my proof

Let m , n and k be positive integers. Prove that v_p(mn)=v_p(m)+v_p(n) , (we define v_p(n) as the greatest ineger r such that p^r  divides n.

My proof is :

  Let v_p(mn)=p^r that divides mn , if (p^r, m)=1 , then p^r must divides n , then  v_p(m)=0 , v_p(n)=p^r , so v_p(mn)=v_p(m)+v_p(n), vice versa

  If (p^r, m) not equal to 1 , and so is (p^r, n) , there exists i , k such that  p^r=p^i*p^k   (p^i,m)=p^i , (p^k,n)=p^k,then r=i+k
v_p(mn)=v_p(m)+v_p(n). ,


CHeck my proof , I am unsure of the second part.


Numbers are the essence of the Universe

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#2 2007-02-02 10:17:14

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Check my proof

Haven't read your proof yet, but here is mine:




Note that this is not exactly the same as your statement, but that they are in fact equivalent.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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