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Get the bases! (Posted on 2018-10-17) Difficulty: 3 of 5
The number 1987 can be written as a three digit number xyz in some base b.
If x + y + z = 1 + 9 + 8 + 7=25, determine all possible values of x, y, z, b.

Source: 1987 Canadian Mathematical Olympiad.

See The Solution Submitted by Ady TZIDON    
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computer soln | Comment 1 of 5
Well, the only shortcut I saw was that 13<b<45 to give a three digit number with x non-zero. I tried by hand up to b=15 and then fell back to the computer. Base 19 held the answer: 

  5  9 11 base   19 adds to 25

I am painfully aware that these kids had no computers - so how did they limit the choices? 

        program zzz

         do 2 i=13,45

         ix=1987/i**2

         is=1987-ix*i**2

         iz=is-is/i*i

2        if(ix+is/i+iz.eq.25)print*,ix,is/i,iz,'base',i,' adds to 25'

        end

rabbit-3:~ lord$ zzz

  5  9 11 base   19 adds to 25
note, for x=0, looping up to base 2000, these work too:

rabbit-3:~ lord$ zzz

  0 18  7 base  110 adds to 25

  0  9 16 base  219 adds to 25

  0  6 19 base  328 adds to 25

  0  3 22 base  655 adds to 25

  0  2 23 base  982 adds to 25

 0  1 24 base 1963 adds to 25


Edited on October 17, 2018, 4:39 pm
  Posted by Steven Lord on 2018-10-17 10:41:26

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