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Extraterrestrial primes (Posted on 2004-01-19) Difficulty: 3 of 5
An earthling with a superpower telescope observed a chalkboard on a distant planet. On it were some mathematical statements. After months of translating, he successfully translated all the words and digits. Unfortunately, due to the complexity of the language, he couldn’t figure out the exact number of ones in each number. All he knows is that they each have at least 2 ones and the first number (but not necessarily the second) has an even number of ones. Other than the ones, the only other digit is a single two. The following is the furthest he could translate it:

1…1 [with an even number of ones] is a prime number
1…12 is a prime number

Assuming both numbers use the same base number, prove that someone or something made a mistake.

See The Solution Submitted by Tristan    
Rating: 3.4000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution re: It looks to me like ... | Comment 2 of 7 |
(In reply to It looks to me like ... by Richard)

The first number is never prime. As long as it's written in at least base 3, it's divisible by 101010101... times 11;

For example, if it was base 5, it would be (6*1)+(6*25)+(6*625)... in base 10.

The second number would be represented by (11*10)+2, and since the first number is 1 more than the second, one of them must be divisible by two. Even times odd plus even is still even and divisible by 2 so it's not prime either.

For example, 112 in base 5 is (6*5)+2, or 32, in base 6 it's (7*6)+2 or 44.
  Posted by Gamer on 2004-01-19 14:08:48

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