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Elegant Exponent Exposition II (Posted on 2010-12-04) Difficulty: 3 of 5
Each of the capital letters in bold denotes a different base ten digit from 1 to 9 in this cryptarithmetic expression:

                                         AB/C + DE/F + GH/I

(I) Determine the respective smallest and the largest positive integer that this expression can assume.

(II) Determine the respective smallest and the largest prime number that this expression can assume. What are the respective smallest and the largest palindrome assumed by the above expression?

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer solution for integers and primes | Comment 1 of 3
    5   point 9
   10   S$="123456789":H$=S$
   15   Highval=0:Lowval=999999999999999:Highprime=0:Lowprime=99999999999999
   16   Highpal=0:Lowpal=999999999999999
   20   repeat
   30        gosub *Permute(&S$)
   40        Abc=val(mid(S$,1,1))^(val(mid(S$,2,1))/val(mid(S$,3,1)))
   41        Def=val(mid(S$,4,1))^(val(mid(S$,5,1))/val(mid(S$,6,1)))
   42        Ghi=val(mid(S$,7,1))^(val(mid(S$,8,1))/val(mid(S$,9,1)))
   50        Sum=Abc+Def+Ghi
   60        if abs(Sum-int(Sum+0.5))<0.000000000000001 then
   65            :Sum=int(Sum+0.5)
   70            :if Sum>Highval then Highval=Sum:Highstr=S$:endif
   75            :if Sum<Lowval then Lowval=Sum:Lowstr=S$:endif
   80            :if prmdiv(Sum)=Sum and Sum>Highprime then Highprime=Sum:Highpr
str=S$:endif
   85            :if prmdiv(Sum)=Sum and Sum<Lowprime then Lowprime=Sum:Lowprstr
=S$:endif
   90              :if prmdiv(Sum)=0 then print "***";Sum;S$:endif
  400   until S$=H$
  700   print Lowstr,Lowval:print Highstr,Highval:print
  710   print Lowprstr,Lowprime:print Highprstr,Highprime:print
  799   end
  800
  900   *Permute(&A$)

finds:

157263948        8
472563891        134217881

157293846        13
542781963        5764907

Translated that's:

1^(5/7)+2^(6/3)+9^(4/8) =       8
4^(7/2)+5^(6/3)+8^(9/1) =       134,217,881

1^(5/7)+2^(9/3)+8^(4/6) =       13
5^(4/2)+7^(8/1)+9^(6/3)  =      5,764,907

Does 8 count as a palindrome?


  Posted by Charlie on 2010-12-04 18:26:37
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