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Question Number 142880 by Dwaipayan Shikari last updated on 06/Jun/21
Provethatϕ(n)=n∏k(1−1pk)ϕ(n):Eulertotientfunction
Answered by Snail last updated on 06/Jun/21
Iamconsideringthatuknowϕ(n)ismultiplicativefunctioni.eϕ(ab)=ϕ(a)ϕ(b)..whereaandbarecoprimeLetp1.............pkaredistinctprimdfactorsofnn=p1k1.........pkkr[herer=kask=numberofterms]ϕ(n)=ϕ(p1k1.....pkkr)=ϕ(p1k1)......ϕ(pkkr)....(1)Nowthenumberofintegerswhicharenotcoprimetopk...(p.1),(p.2),(p.3),........(p.pk−1)ϕ(pk)=numbdrofintegerscoprimdtopkand<pk=pk−pk−1=pk(1−1p)Nowusingthisin(1)=p1k1(1−1p1)p2k2(1−1p2)...pkkr(1−1pr)=p1k1p2k2.....prkr(1−1p1).....(1−1pr)=n(1−1p1)(1−1p2)....(1−1pk)Proved....
Commented by Dwaipayan Shikari last updated on 07/Jun/21
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