; TeX output 2001.10.11:2000 "ZE⍑q"V cmbx10MathT124ProblemSet1uTq1.UUK`y cmr10First,weshowthatthebinomialcoGecientisaninteger. qClaim:UUF*orxed b> cmmi10nand0 !", cmsy10kn,UUb u cmex10 0er cmmi7n⾍-k*bֲisUUaninteger.q ': cmti10Pr}'oof.mByinduction.Clearlyb JٓR cmr71⾍J0b2=b u1⾍u1b]=1.SuppGosethatb Jn⾍FAkȟbisanintegerfor0k@n.Nowqb rȴn+1⾍#O0-Ib4า=b \on+1⾍\on+1mb_=1UUandfor0in8 1<"^<$Cjn 卑DƽiI^SX+8^<$qn 卑 Ti8+1@'^(c=<$l;n!Kw fe 7 (֍i!(n8 i 1)!>^<$Oi1Gdw fe l (֍n8 i`*+<$A|1lw fe ӟ (֍i8+1J^ _=<$(n8+1)!Kw fe ?r (֍(i8+1)!(n i)!GgK=^<$ #n8+1 卑jߵi8+1!]^(y;+qwhichUUbytheinductivehypGothesisisthesumoftwointegers.qThisprovestheclaim.qSince?pisprime,zTitissucienttoshowthatthereisnofactorofpinthedenominator.0Byqassumption,r<epsorG!doesnotcontainafactorofp.FTSimilarly*,1er8impliesthatp rqp8 1