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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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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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