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The sequence $10000, 121, 100, 31, 24, n, 20$ represents a number $x$ with respect to different bases. What is the missing number, $n$?

This is from my elementary computer aptitude paper. Is there any way to solve this quickly?

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You can interpolate between $24$ and $20$ to obtain $22$. Is that the kind of trick you're looking for? –  Yuval Filmus Apr 3 '11 at 18:09
    
@Yuval Filmus:I don't get you .. could you please be more explicit? –  Quixotic Apr 3 '11 at 18:15
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Yuval and I are both assuming the bases are in ascending order with a difference of $1$. His interpolation is essentially an informal version of my answer. A little algebra beyond my answer shows the number is 16 and the bases are 2,3,4,5,6,?,8 –  Ross Millikan Apr 3 '11 at 18:32

3 Answers 3

If the base in the last term is $b$, the number is $2b$. The missing term is then in base $b-1$. Expressed in base $b-1$ the integer is $2(b-1)+2=22$ The third to last term shows that $b-2 \ge 5$ so we are safe from a carry.

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The bases are 2,3,4,5,6,7,8, expressing values 1*16=16, 9 + 6 + 1 = 16, 1*16=16, 15 + 1 = 16, 12 + 4 = 16, 7*x + y = 16, and 2*8 = 16. x and y are then 2 and 2.

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Assuming the bases are in ascending order with a difference of 1, and the first one is in base $b$, then $$b^4=(b+2)^2$$ and so $$b^2=b+2$$ from which we can conclude that $b=2$ and the number is 16, which in base 7 is represented as 22.

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