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Numbers whose factorial has '84' as its final two digits before the trailing zeros.
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%I #17 Jul 02 2022 17:48:05

%S 25,26,42,55,103,136,140,147,158,169,231,261,265,272,281,333,344,356,

%T 380,389,410,460,472,486,503,565,582,595,598,629,662,674,677,683,694,

%U 700,701,712,742,763,777,789,839

%N Numbers whose factorial has '84' as its final two digits before the trailing zeros.

%H Charles R Greathouse IV, <a href="/A045567/b045567.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>

%t Select[Range[5,850],Take[Flatten[Most[Split[IntegerDigits[#!]]]],-2]=={8,4}&] (* _Harvey P. Dale_, Jul 02 2022 *)

%o (PARI) last(n,k)=if(n*8\9-1<k, n=n!; return(n/10^valuation(n,10)%10^k)); my(m=Mod(1,5^k)); for(i=2,n,m*=i/10^valuation(i,5));lift(chinese(m, Mod(0,2^k))) \\ Gives the last k decimal digits of n!.

%o is(n)=last(n,2)==84 \\ _Charles R Greathouse IV_, Oct 22 2014

%o (PARI) is(n)=my(m=Mod(1,25)); for(i=2,n, m*=i/10^valuation(i,5)); m==84 \\ _Charles R Greathouse IV_, Oct 22 2014

%o (PARI) v=List();m=Mod(1,25);for(n=2,1e6,m*=n/10^valuation(n,5);if(m==84,listput(v,n);if(#v==100, return(Vec(v))))) \\ _Charles R Greathouse IV_, Oct 22 2014

%K nonn,base

%O 1,1

%A _Jeff Burch_