Number Theory

Number Bases

Number Bases: Level 5 Challenges

         

The first expedition to a distant planet found only the ruins of a civilization. The explorers were able to translate the "extraterrestrial" equation they found there as follows:

5x250x+125=0. Therefore, x is equal to 5 or 8. 5x^2 -50x +125 = 0. \text{ Therefore, } x \text{ is equal to } 5 \text{ or } 8.

This was an extremely strange result. The value x=5x=5 seemed satisfactory, but x=8x=8 required some explanation. If their number system were similar to ours, how many fingers would you say the inhabitants in that planet had?

Make the assumption that if the inhabitants had nn fingers, then they work in base nn.

What is the 50th smallest positive integer that can be written as the sum of one or more distinct powers of 3 with non-negative integer exponents?

In the infinite series 19+199++110n1+\dfrac{1}{9}+\dfrac{1}{99}+\cdots + \dfrac{1}{10^n-1}+\cdotsat what place after the decimal point does the second 11 occur?


Details and Assumptions\text{Details and Assumptions}

The first 11 occurs at the first place after the decimal point.

Doge has become the king of a huge empire and wants to show the world that he is really intelligent. He is so over ambitious that his first action as king is to name the currency after himself: DogeCoin. He wants to issue dd denominations of coins so that using no more than 3 coins, citizens can pay any amount from 1 DogeCoin to 36 DogeCoin in exact change.

Find the minimum value of dd and all the corresponding denominations of DogeCoin. Submit your answer as the sum of these denominations.

Details and Assumptions:

  • As an explicit example, if you get the denominations as {10,12,15} \{10, 12, 15\}, then input the answer as 10+12+15=3710+12+15=37 .

There exists a unique 10-digit number abcdefghij\overline{abcdefghij} which contains each of the digits 0, 1, 2, \dots, 9 exactly once, such that for each kk, 1k101 \le k \le 10, the number formed by the first kk digits of abcdefghij\overline{abcdefghij} is divisible by kk. For example, for k=4k = 4, the number abcd\overline{abcd} is divisible by 4.

Find the three-digit number abc\overline{abc}.

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